Commuting with Autonomous Vehicles: A Branch and Cut Algorithm with Redundant Modeling
CCommuting with Autonomous Vehicles: A Branchand Cut Algorithm with Redundant Modeling
Mohd. Hafiz Hasan
University of Michigan, Ann Arbor, Michigan 48105, USA, [email protected]
Pascal Van Hentenryck
Georgia Institute of Technology, Atlanta, Georgia 30332, USA, [email protected]
This paper studies the benefits of autonomous vehicles in ride-sharing platforms dedicated to serving com-muting needs. It considers the Commute Trip Sharing Problem with Autonomous Vehicles (CTSPAV), theoptimization problem faced by a reservation-based platform that receives daily commute-trip requests andserves them with a fleet of autonomous vehicles. The CTSPAV can be viewed as a special case of the Dial-A-Ride Problem (DARP). However, this paper recognizes that commuting trips exhibit special spatial andtemporal properties that can be exploited in a branch and cut algorithm that leverages a redundant model-ing approach. In particular, the branch and cut algorithm relies on a MIP formulation that schedules miniroutes representing inbound or outbound trips. This formulation is effective in finding high-quality solutionsquickly but its relaxation is relatively weak. To remedy this limitation, the mini-route MIP is complementedby a DARP formulation which is not as effective in obtaining primal solutions but has a stronger relaxation.A column-generation procedure to compute the DARP relaxation is thus executed in parallel with the corebranch and cut algorithm and asynchronously produces a stream of increasingly stronger lower bounds. Thebenefits of the proposed approach are demonstrated by comparing it with another, more traditional, exactbranch and cut procedure and a heuristic method based on mini routes.The methodological contribution is complemented by a comprehensive analysis of a CTSPAV platform forreducing vehicle counts, travel distances, and congestion. In particular, the case study for a medium-sizedcity reveals that a CTSPAV platform can reduce daily vehicle counts by a staggering 92% and decreasevehicles miles by 30%. The platform also significantly reduces congestion, measured as the number of vehicleson the road per unit time, by 60% during peak times. These benefits, however, come at the expense ofintroducing empty miles. Hence the paper also highlights the tradeoffs between future ride-sharing andcar-pooling platforms.
Key words : autonomous vehicles, shared commuting, branch-and-cut, column generation
1. Introduction
This work is the culmination of a four-year study on the benefits of ride-sharing and car-poolingplatforms for serving commuting needs. It was originally motivated by the desire to relieve parkingpressure in the city of Ann Arbor, Michigan. Parking structures are expensive and are often locatedin prime locations for the convenience of commuters. In Ann Arbor, the parking pressure wasprimarily caused by commuters to the University of Michigan, the city’s largest employer withmore than 50,000 employees. a r X i v : . [ m a t h . O C ] J a n asan and Van Hentenryck: Commuting with Autonomous Vehicles Detailed information about the commuting patterns of these employees was gathered by recordingtrip data from approximately 15,000 drivers who use the 15 university-operated parking structureslocated in the downtown area over the month of April 2017. The data consisted of the exact arrivaland departure times of every commuter to the parking structures, which was then joined with theprecise locations of the parking structures and the home addresses of every commuter to reconstructtheir daily trips. The dataset revealed several intriguing temporal and spatial characteristics. First,the peak arrival and departure times, which are depicted in Figure 1 for the weekdays of the busiestweek, coincide with the typical peak commuting hours. Second, the strong consistency of the tripschedules was seen as a significant opportunity for car-pooling and ride-sharing platforms. Third,the commuting destinations (the parking structures) are located within close vicinity of each otherin the downtown area (as they are university-owned structures), whereas the commuting origins(the commuter homes) are located in the neighborhoods surrounding the downtown area, as well asin Ann Arbor’s neighboring towns. This spatial structure, which is quite typical of many Americancities, was also seen as an opportunity for trip-sharing platforms.With this in mind, Hasan et al. (2020) introduced the Commute Trip Sharing Problem (CTSP) toformalize the key optimization problem faced by a car-pooling platform that would serve commutetrips. More precisely, the CTSP conceptualizes the platform as a reservation-based system thatreceives the commute-trip requests—each consisting of a trip request to the workplace (inboundtrip) and another to return back home (outbound trip)—ahead of time (e.g., the day ahead or themorning of each day). Each trip request includes small time windows for its departure and arrival,and each rider is guaranteed not to spend more than R % of her direct trip in commuting time.The CTSP was tailored to scenarios where: (1) The commuters travel to a common/centralizedlocation, e.g., the commute trips of the employees of a corporate or university campus, or (2)The commuters live in a common/centralized location, e.g., the commute trips originating from aresidential neighborhood or an apartment complex. These scenarios were inspired by the spatio-temporal structure observed in the Ann Arbor commute-trip dataset described earlier.To implement such a platform and address the complexity of dealing with the massive volumeof the trips from the dataset, Hasan et al. (2020) applied a two-stage approach:1. it first clusters commuters into artificial neighborhoods based on the spatial proximity of theirhome locations, using an unsupervised machine-learning algorithm;2. it then finds optimal routes for the commuters within each cluster.Figure 2 provides an overview of the resulting clusters within Ann Arbor’s city limits: it displaysthe convex hulls of the neighborhoods, as well as the convex hull of the centrally located parkingstructures. The optimization problem in step 2 is the CTSP: each day, its goal is to use privatevehicles owned by the commuters, select the set of drivers for the inbound and outbound routes of asan and Van Hentenryck: Commuting with Autonomous Vehicles T r i p c o un t Hour of day
Arrival Departure
Monday Tuesday Wednesday Thursday Friday
Figure 1 Distribution of Arrival and Departure Times Over Week 2 of April 2017Figure 2 Convex Hulls of Artificial Neighborhoods Resulting from Clustering Algorithm the vehicles, and design the routes in order to minimize the number of vehicles utilized, and hencethe parking pressure. Solutions to the CTSP were shown to reduce daily vehicle usage for the AnnArbor dataset by up to 57%. asan and Van Hentenryck:
Commuting with Autonomous Vehicles Despite this significant potential, the results also highlighted several factors limiting furtherreductions in vehicle counts. They included (1) the nature of the CTSP routes that are typicallyshort and (2) the necessity to synchronize the inbound and outbound routes since they must beperformed by the same set of drivers. Indeed, as the drivers in the CTSP are selected from the setof commuters themselves, each route must begin and end at the origin and destination of its driver.This book-ending requirement subjects the total duration of the route to the temporal constraintsof the driver, restricting its length and consequently its ability to serve more trips. This, combinedwith the necessity of selecting an identical set of drivers for the inbound and outbound routes,limits the flexibility of the routes that can be generated and used in a CTSP routing plan.The Commute Trip Sharing Problem with Autonomous Vehicles (CTSPAV) considered in thispaper was originally proposed by Hasan and Van Hentenryck (in press 2021): its goal was toovercome these shortcomings by leveraging Autonomous Vehicle (AV) technology that is lurking inthe horizon. By removing driver-related constraints, the CTSPAV was anticipated to allow the AVroutes to be significantly longer than the CTSP routes. While these longer routes would significantlyincrease the number of commute trips that can be covered by each AV on any day, the algorithmiccomplexity for finding them was also expected to increase significantly. Hasan and Van Hentenryck(in press 2021) therefore proposed a column-generation solution procedure, dubbed the CTSPAVprocedure, that is a departure from the classical column-generation approach for solving typicalVehicle Routing Problems (VRPs). The latter typically entails solving a set-partitioning/coveringmaster problem that ensures that each customer is served, and a pricing subproblem that searchesfor feasible routes that depart from and return to a depot and have negative reduced costs. TheCTSPAV procedure circumvents the anticipated complexity of searching for the long AV routes inthe pricing subproblem by shifting some of the burden to the master problem and exploiting thespatio-temporal structure of the dataset . It uses a pricing subproblem that only searches for feasible“mini” routes with negative reduced costs instead. The mini routes are short by construction:each covers only inbound or outbound trips exclusively, and each has distinct pickup, transit, anddrop-off phases during which it first visits trip origins, then travels from an origin to a destination,and finally visits destinations. These three phases are naturally encountered by each vehicle as ittravels from a residential neighborhood to the workplace in the morning to serve inbound trips,and vice versa in the evening to serve outbound trips.In order for these mini routes to be feasible, they must visit each location within a specifiedtime window, ensure that the time spent of the vehicle by each rider does not exceed a specifiedlimit, and cannot exceed the vehicle capacity. In other words, they must satisfy time-window,ride-duration, and vehicle-capacity constraints. Furthermore, they must also satisfy pairing andprecedence constraints, which require a route visiting the origin of trip to also visit its destination asan and Van Hentenryck:
Commuting with Autonomous Vehicles in the correct order. The master problem of the CTSPAV procedure is then responsible for stitchingor chaining together the feasible mini routes to form longer AV routes that begin and end at adepot. In addition to ensuring that each trip is covered, the master problem must also select miniroutes that are temporally compatible with each other, i.e., it needs to ensure that it is possible totravel from the last destination of one mini route to the first origin of another without violatingthe temporal constraints of the selected mini routes. All of this is done in service of a lexicographicobjective function that first minimizes the number of formed AV routes (i.e., the vehicle count ifeach route is assigned to an AV) and then minimizes their total travel distance.Since the routes of the CTSPAV satisfy time-window, ride-duration, capacity, pairing, and prece-dence constraints which are identical to those for the Dial-A-Ride Problem (DARP) (Cordeau andLaporte 2003a, 2007), the CTSPAV can be seen as a special version of the DARP that servesinbound-outbound trip pairs using AVs. In fact, any DARP algorithm can be used to solve theCTSPAV. Hasan and Van Hentenryck (in press 2021) explored this possibility as well by investigat-ing a DARP procedure for solving the CTSPAV. The procedure can be thought of as a model-drivenapproach that borrows heavily from an algorithm for the DARP proposed by Gschwind and Irnich(2015), as it relies on the classical column-generation approach but uses a novel, label-settingdynamic program to solve its pricing subproblem. Hasan and Van Hentenryck (in press 2021) dis-covered that, while the complexity of discovering the long AV routes in its pricing subproblemseverely hampered the algorithm ability to find strong integer solutions within a time-constrainedsetting, the DARP model also produced superior primal lower bounds for the primary objective.On the other hand, the CTSPAV procedure produces stronger integer solutions within a similartime-constrained setting, but it does so at the expense of generating weaker lower bounds.This paper aims at addressing these limitations with two goals in mind:1. to propose an exact algorithm for the CTSPAV;2. to provide a conclusive and comprehensive analysis of the potential of the CTSPAV for reduc-ing vehicle counts, travel distances, and congestion.To meet the first goal, the paper presents an exact algorithm that improves upon the CTSPAVprocedure of Hasan and Van Hentenryck (in press 2021) by combining the insights from bothapproaches in a redundant modeling framework (Liberti 2004, Ruiz and Grossmann 2011). Theproposed algorithm leverages the best characteristics of the CTSPAV and DARP procedures, i.e.,the former’s capability of producing strong integer solutions and the latter’s ability of generatingstrong primal lower bounds. More specifically, the paper describes a branch-and-cut procedurewhich is capable of solving medium-sized CTSPAV instances exactly, unlike the CTSPAV proce-dure of Hasan and Van Hentenryck (in press 2021). This procedure is then compared against abranch-and-cut procedure using other families of valid inequalities, as well as against the CTSPAV asan and Van Hentenryck: Commuting with Autonomous Vehicles procedure of Hasan and Van Hentenryck (in press 2021) for problem instances derived from theAnn Arbor commute-trip dataset. With the exact CTSPAV algorithm available, the paper can thenperform a systematic analysis of the CTSPAV potential in reducing vehicle counts, travel distance,and congestion. Moreover, the paper can contrast the existing situation where commuters drivemostly alone with car-pooling and automomous ride-sharing platforms, highlighting the varioustrade-offs on a real case study.The methodological contribution of this paper is to propose a branch-and-cut algorithm for solvingthe CTSPAV exploiting a novel dual-modeling technique. The branch and cut algorithm solves amathematical model that exploits the spatio-temporal structure of the data, making it conduciveto finding high-quality solutions quickly. But the branch and cut algorithm also uses anothermathematical model for the same problem to generate valid inequalities that are separated bya column-generation procedure and produce strong lower bounds. The paper demonstrates thebenefits of this dual-modeling approach through a comparison with a dedicated branch-and-cutprocedure based on well-established families of valid inequalities, and with the heuristic column-generation procedure of Hasan and Van Hentenryck (in press 2021). The proposed exact branch andcut procedure is also embedded into a end-to-end approach combining clustering and optimizationto solve large-scale, real-world instances of the CTSPAV.The methodology ontribution is completemented by a case study that provides unique insightson the potential benefits of ride sharing and autonomous vehicles for serving the commuting needsof many cities around the world. The case study demonstrates that a ride-sharing platform basedon autonomous vehicles can provide substantial reductions in vehicle counts and congestion, as wellas improvements in travel miles. In addition, the paper contrasts, for the first time, the potentialbenefits and drawbacks of car-pooling and ride-sharing platforms along those dimensions.The rest of this paper is organized as follows. Section 2 briefly discusses related work. Section3 introduces the terminologies and assumptions used throughout the work. Section 4 describesthe clustering algorithm. Section 5 specifies the CTSPAV model and describes an algorithm forenumerating mini routes. Section 6 provides an overview of the branch-and-cut algorithm and coversthe different families of valid inequalities considered in this work together with the heuristics usedto separate them. Section 7 outlines how the algorithm is evaluated and presents the computationalresults. Section 8 documents the insights obtained on the case study. Finally, Section 9 providessome concluding remarks.
2. Related Work
The Vehicle Routing Problem with Time Windows (VRPTW) is perhaps the most well-studiedvariant of VRPs; It seeks an optimal routing plan that consists of a set of minimum cost routes, asan and Van Hentenryck:
Commuting with Autonomous Vehicles each departing and returning to a designated depot, to service a set of customers. Each customerhas a capacity demand and a time window specifying allowable service times, therefore the planmust ensure every customer is served exactly once within their time windows while not exceedingthe capacity of the vehicles utilized, i.e., its routes must satisfy time-window and vehicle-capacityconstraints. The problem is well-known to be NP-hard as finding a feasible solution to the versionof the problem with a fixed vehicle count has been shown to be NP-complete by Savelsbergh (1985).Nevertheless, numerous approaches ranging from heuristics to exact methods have been proposedfor the problem, and they have been comprehensively reviewed by Cordeau et al. (2002). TheVRPTW was generalized to the Pickup and Delivery Problem with Time Windows (PDPTW)by Dumas et al. (1991) to model services that first pick up and then deliver merchandise withinspecified time windows. The routes of the problem therefore need to satisfy pairing and precedenceconstraints in addition to time-window and vehicle-capacity. The former two require that each routevisit a pair of locations associated with each customer in a specific order, the first representing apickup location and the second representing a delivery location. The PDPTW was then generalizedto the DARP which is used to model door-to-door transportation services for the disabled or theelderly. The ride duration becomes a critical factor for ensuring the quality of these services asthey are now transporting humans. Therefore the DARP introduces ride-duration constraints tothe PDPTW, which limit the time elapsed between every pair of pickup and delivery location toensure that the customers are not spending excessive amounts of time on the vehicle. The variousalgorithms and techniques that have been proposed for the DARP have been reviewed by Cordeauand Laporte (2003a, 2007).Of the many solution approaches that have been proposed for the different variants of the VRP,column generation is perhaps to most popular due to its ability to generate strong lower bounds tothe problem objective and due to its elegance of only considering a subset of feasible routes that canimprove the objective function. The typical column-generation approach for solving VRPs beginswith the application of the Dantzig-Wolfe decomposition (Dantzig and Wolfe 1960) on an edge-flowformulation of the problem to produce a master problem and a pricing subproblem. The masterproblem typically solves a set-partitioning/covering problem on a set of feasible routes to ensureevery customer is served, whereas the pricing subproblem searches for new feasible routes to beadded to the set. The latter problem uses the duals of the linear relaxation of the master problem toidentify new routes with negative reduced costs, and it is typically cast as a Shortest Path Problemwith Resource Constraints (SPPRC), a class of problems that has been extensively reviewed byIrnich and Desaulniers (2005). The SPPRC seeks a route with minimum cost, and the feasibilityof the discovered route is guaranteed through the enforcement of numerous resource constraintsthat model the route-feasibility constraints. Some of the approaches that have been used to solve asan and Van Hentenryck: Commuting with Autonomous Vehicles these SPPRCs include Lagrangian relaxation (Beasley and Christofides 1989, Bornd¨orfer et al.2001), constraint programming (Rousseau et al. 2004), heuristics (Desaulniers et al. 2008), andcutting planes (Drexl 2013), but perhaps the most popular approach uses dynamic programming,e.g., the generalized label-setting algorithm for multiple resource constraints by Desrochers (1988).Examples of successful applications of column generation on the different VRP variants includeDesrosiers et al. (1984), Desrochers et al. (1992) for the VRPTW, Dumas et al. (1991), Ropke andCordeau (2009) for the PDPTW, and Gschwind and Irnich (2015) for the DARP.Another common approach for solving routing problems is the polyhedral approach which gen-erates cutting planes to progressively “trim” the convex hull defining the feasible region of theproblem’s linear relaxation. Its application on VRPs traces its roots back to the seminal work byDantzig et al. (1954) for solving the Traveling Salesman Problem (TSP). Their procedure uses anedge-flow formulation of the problem which is iteratively solved to identify subtours which breakthe feasibility of the solution. A family of valid inequalities, commonly referred to now as the DFJsubtour elimination constraints (SECs), are then progressively introduced to prevent generation ofthe subtours in subsequent solutions. Gr¨otschel and Padberg (1975) later proved that the DFJ SECsinduce facets of the polytope of the convex hull of the feasible solutions, which explained why theywere so effective at strengthening the linear-programming (LP) bound, while Padberg and Rinaldi(1990) proposed an exact algorithm for separating the inequalities. In a similar vein, many otherworks have focused on identifying facet-defining inequalities together with algorithms/heuristics forseparating them, e.g., D + k and D − k inequalities for the TSP by Gr¨otschel and Padberg (1985), pre-decessor and successor inequalities for the Precedence-Constrained Asymmetric TSP (PCATSP)by Balas et al. (1995), tournament and generalized tournament constraints for the AsymmetricTSP with Time Windows (ATSPTW) by Ascheuer et al. (2000), and 2-path cuts for the VRPTWby Kohl et al. (1999). Most approaches to routing problems embed cutting-plane generation withinthe classical branch-and-bound framework for solving mixed-integer programs (MIPs) to producea more sophisticated branch-and-cut procedure, whereby heuristics for separating violated validinequalities are executed on the solution of the LP relaxation that is obtained in the boundingphase of each tree node. The separated inequalities are then introduced into the problem formu-lation to strengthen the LP bound of the procedure. The proposed branch-and-cut algorithmstypically begin with an edge-flow formulation and then introduce numerous existing and/or newfamilies of valid inequalities that are tailored specifically for the type of routing problem beingsolved. Examples of these branch-and-cut algorithms include Padberg and Rinaldi (1991) for theTSP, Fischetti and Toth (1997) for the Asymmetric TSP (ATSP), Ruland and Rodin (1997) forthe Pickup and Delivery Problem (PDP), Ascheuer et al. (2001) for the ATSPTW, Naddef and asan and Van Hentenryck: Commuting with Autonomous Vehicles Rinaldi (2001) for the Capacitated VRP (CVRP), Bard et al. (2002), Kallehauge et al. (2007) forthe VRPTW, and Cordeau (2006) for the DARP.The prevalence of large-scale datasets of real-world trips, e.g., the New York City (NYC) Taxi &Limousine Commission (TLC) trip record data (NYC Taxi & Limousine Commission 2020) whichstores trip information of more than one billion taxi rides in NYC, combined with the growingawareness and concern for the sustainability of passenger mobility systems have increased attentiontowards the optimization of car-pooling and ridesharing services. For instance, Santi et al. (2014)formalized the notion of shareability networks as a tool to quantify the ridesharing potential ofthe trips from the TLC dataset, while Alonso-Mora et al. (2017) proposed an anytime optimalalgorithm that utilizes shareability graphs to optimize ridesharing for on-demand trip requestsextracted from the TLC dataset. Studies involving other real-world datasets include Baldacci et al.(2004) who proposed a Lagrangian column-generation method to optimize the Car-Pooling Problem(CPP) for commuting trips to a research institution in Italy and Agatz et al. (2011) who used graphmatching within a rolling-horizon framework to optimize ridesharing for real-time, non-recurringtrips from metro Atlanta. Classifications of the different variants of shared mobility problemstogether with reviews of the proposed optimization approaches for them are provided by Agatzet al. (2012) and Mourad et al. (2019). The impending arrival of fully autonomous vehicles has alsospurred a growing interest in the potential of Shared Autonomous Vehicle (SAV) services, due tothe perceived benefits that are afforded by this new mode of transportation, be it reducing traffic(Martinez and Viegas 2017, Alazzawi et al. 2018, Salazar et al. 2018), increasing road capacity(Friedrich 2015, Tientrakool et al. 2011, Talebpour and Mahmassani 2016, Mena-Oreja et al. 2018,Olia et al. 2018), or decreasing parking demand (Zhang et al. 2015, Dia and Javanshour 2017,Zhang and Guhathakurta 2017). Narayanan et al. (2020), which reviewed the numerous potentialimpacts of SAV services to society and the environment, also suggested classifying them as eitheron-demand or reservation-based systems, with the former being tailored for dynamic trips whoserequests are made in real time and the latter for recurring trips whose requests are made way inadvance. Several optimization approaches have also been proposed for conceptual systems of eachtype. For example, Farhan and Chen (2018) proposed a three-step approach—which clusters triprequests from discretized time intervals by assigning them to their nearest vehicles and then solvingthe requests for each cluster as a VRPTW—to optimize a fleet of SAVs for on-demand trips, whileMa et al. (2017) proposed an LP approach to optimize vehicle sharing of a fleet of SAVs for triprequests that are known ahead of time.The work on the CTSPAV traces its roots back to the authors’ initial desire to solve the parkingproblem in downtown Ann Arbor, Michigan, that was partly caused by the massive infusion oftrips from the thousands of commuters driving to the University of Michigan campus daily. Having asan and Van Hentenryck:
Commuting with Autonomous Vehicles access to a large-scale, high-fidelity dataset of these commute trips, they wanted to investigate thevehicle reduction potential of an optimized car-pooling or ridesharing platform. Hasan et al. (2018)began by investigating the performance of several car-pooling and car-sharing models, each withdifferent driver and passenger matching constraints, and discovered that the model that requiresthe commuters to adopt different roles and to ride with different passengers and drivers dailyhad the best vehicle reduction potential. In other words, the flexibility in driving and sharingpreferences is critical to maximizing trip shareablity. In (Hasan et al. 2020), the best performingcar-pooling model was refined and subsequently formalized as the CTSP, a model that maximizestrip sharing while selecting an identical set of drivers for the inbound and outbound routes fromthe set of commuters on a daily basis. Two exact algorithms were proposed: the first exhaustivelyenumerates feasible routes before their selection is optimized with a MIP, while the second usescolumn generation to search for feasible routes on demand within a branch-and-price framework.Subsequent application of the algorithms on the commute-trip dataset revealed an ability to reducedaily vehicle counts by more than 50%. Hasan and Van Hentenryck (2020) then proposed a methodto handle potential uncertainties in the trip schedules of the CTSP by incorporating a randomized,scenario-sampling technique within a two-stage optimization approach. The method was shownto be capable of producing routing plans that are robust to changes in trip schedules, but theincrease in robustness comes at the price of an increase in vehicle utilization. A method to properlyevaluate this trade-off was then proposed. The CTSPAV was formally conceptualized in Hasanand Van Hentenryck (in press 2021) to address a key shortcoming of the CTSP—its short routeswhich limited the potential to further reduce daily vehicle counts—through the utilization of a SAVplatform. The work explored two methods for optimizing its routes: (1) an approach which usescolumn generation to search for mini routes which are then assembled in a master problem, and(2) an approach which relied on a more classical column-generation technique originally conceivedfor the DARP. They discovered that each method had complementary performance trade-offs,with the former being able to produce stronger integer solutions and the latter being able togenerate stronger lower bounds. All of these earlier works have culminated into this study whichhopes to develop an algorithm that melds together both approaches proposed from Hasan andVan Hentenryck (in press 2021) in order to leverage their unique strengths in effectively solvingthe CTSPAV. Accomplishing this goal uniquely positions this work to glean additional insightsinto the strengths and weaknesses of an optimized SAV platform relative to car-pooling platformsthat uses conventional vehicles for maximizing large-scale ridesharing of commute trips.
3. Preliminaries
This section introduces the main concepts used throughout this paper: trips, mini routes, and AVroutes. It also describes the constraints that mini routes and AV routes must satisfy. This work asan and Van Hentenryck:
Commuting with Autonomous Vehicles assumes that a homogeneous fleet of vehicles with capacity K is available to serve all rides, andthat the triangle inequality is satisfied for all travel times. Trips
A trip t = { o, dt, d, at } is a tuple that consists of an origin o , a departure time dt , adestination d , and an arrival time at of a trip request. Every day, a commuter c makes two trips:a trip t + c to the workplace and a return trip t − c back home. These trips are called inbound andoutbound trips respectively. Mini Routes
A mini route r is a sequence of locations that visits each origin and destinationfrom a set of inbound or outbound trips exactly once. Let C r denote the set of riders served in r . A mini route r must respect the vehicle capacity, i.e., |C r | ≤ K , and consists of three phases:a pickup phase where the passengers are picked up, a transit phase where the vehicle travels tothe destination, and a drop-off phase where all the passengers are dropped off. During the pickup(resp., drop-off) phase, the vehicle visits only origins (resp., destinations), whereas it travels froman origin to a destination in the transit phase. For instance, a possible mini route for a car with K = 4 serving trips t = { o , dt , d , at } , t = { o , dt , d , at } , and t = { o , dt , d , at } is r = o → o → o → d → d → d , and its pickup, transit, and drop-off phases are given by o → o → o , o → d , and d → d → d respectively. An inbound mini route r + covers only inbound trips andan outbound mini route r − covers only outbound trips. Definition 1 (Valid Mini Route).
A valid mini route r serving a set of riders C r visits all ofits origins, { o c : c ∈ C r } , before its destinations, { d c : c ∈ C r } , and respects the vehicle capacity, i.e.,it has |C r | ≤ K .Let T i denote the time at which service begins at location i , s i the service duration at i , pred ( i )the location visited just before i , τ ( i,j ) the estimated travel time for the shortest path betweenlocations i and j , and ˙ C r the first commuter served on r . Commuters sharing rides are willing totolerate some inconvenience in terms of deviations to their desired departure and arrival times,as well as in terms of their ride durations compared to their individual, direct trips. Therefore, atime window [ a i , b i ] is constructed around the desired departure times and is associated with eachpickup location i , where a i and b i denote the earliest and latest times at which service may beginat i respectively. Conversely, only an upper bound b j is associated with each drop-off location j as the arrival time at j is implicitly bounded from below by a j = a i + s i + τ ( i,j ) , where i is thecorresponding pickup location for j . On top of that, a duration limit L c is associated with eachrider c to denote her maximum ride duration. Definition 2 (Feasible Mini Route).
A feasible mini route r is valid, has pickup and drop-off times T i ∈ [ a i , b i ] for each location i ∈ r , and ensures the ride duration of each rider c ∈ C r doesnot exceed L c . asan and Van Hentenryck: Commuting with Autonomous Vehicles Determining if a valid mini route r is feasible amounts to solving a feasibility problem defined bythe following constraints on r . a o c ≤ T o c ≤ b o c ∀ c ∈ C r (1) T d c ≤ b d c ∀ c ∈ C r (2) T pred ( o c ) + s pred ( o c ) + τ ( pred ( o c ) ,o c ) ≤ T o c ∀ c ∈ C r \ ˙ C r (3) T pred ( d c ) + s pred ( d c ) + τ ( pred ( d c ) ,d c ) = T d c ∀ c ∈ C r (4) T d c − ( T o c + s o c ) ≤ L c ∀ c ∈ C r (5)Constraints (1) and (2) are time-window constraints for pickup and drop-off locations respectively,while constraints (3) and (4) describe compatibility requirements between pickup/drop-off timesand travel times between consecutive locations along the route. Finally, constraints (5) specifythe ride-duration limit for each rider. Note that constraints (3) allow waiting at pickup locations.Moreover, the service starting times on consecutive locations along r are strictly increasing, whichensures that the route is elementary. Numerous algorithms have been proposed for solving thisfeasibility problem efficiently, e.g. Tang et al. (2010), Haugland and Ho (2010), Firat and Woeg-inger (2011), Gschwind and Irnich (2015). In the following, the Boolean function f easible ( r ) isused to indicate whether mini route r admits a feasible solution to constraints (1)–(5). This workimplements the labeling procedure proposed by Gschwind and Irnich (2015) for this function. AV Routes
An AV route ρ = v s → r → . . . → r k → v t is a sequence of k distinct mini routes thatstarts at a source node v s and ends at a sink node v t , both representing a designated depot. Definition 3 (Feasible AV Route).
A feasible AV route ρ is one that consists of a sequenceof distinct, feasible mini routes and starts and ends at a designated depot.In other words, for ρ to be feasible, each of its mini routes must be valid and satisfy constraints (1)–(5). Let ˙ r denote the first location visited on r and ¨ r denote the last. Each mini route r i (1 ≤ i ≤ k )must also satisfy the following constraints: T v s + τ ( v s , ˙ r ) = T ˙ r (6) T ¨ r i + s ¨ r i + τ (¨ r i , ˙ r i +1 ) ≤ T ˙ r i +1 ∀ i = 1 , . . . , k − T ¨ r k + s ¨ r k + τ (¨ r k ,v t ) = T v t (8)Constraints (6)–(8) describe compatibility requirements between the beginning/ending servicetimes of consecutive mini routes along ρ and the travel times between them. The constraints,together with (3) and (4), enforce strictly increasing starting times for service on all consecutivelocations along ρ , therefore ensuring that ρ is elementary. asan and Van Hentenryck: Commuting with Autonomous Vehicles
4. The Clustering Algorithm
This section describes a clustering algorithm used to decompose the large volume of commute tripsin our case study into smaller, more manageable problem instances. This strategy is congruent withthe conclusion of Agatz et al. (2012) that acknowledges the necessity of effective decompositionapproaches for the computational feasibility of large-scale problems. The idea behind this clusteringapproach is simply to construct artificial neighborhoods within which ridesharing is performedexclusively, and the neighborhoods are constructed by algorithmically grouping up to N commuterstogether based on the spatial proximity of their residential locations. Obviously, this approachprecludes the discovery of a global optimal solution, but it is seen as a practical necessity to ensurethat the problem is computationally tractable.The algorithm proceeds in a fashion that is very similar to the k -means clustering algorithm byLloyd (1982), with the exception that its assignment step limits the number of elements assignedto each cluster by a parameter N to produce groups that are approximately equal in size. Itrepresents each commuter as a point in R whose GPS coordinates are first obtained by geocodingthe commuter home address. In the rest of this section, C denotes the set of point coordinates forevery commuter (i.e., a set of 2D vectors, each storing the 2D coordinates of a commuter home), U the set of coordinates of cluster centers (similarly, a set of 2D vectors, each consisting of the 2Dcoordinates of a cluster center), S ( x ) the Euclidean distance from a point x to the nearest clustercenter, and S ( x , y ) the Euclidean distance between points x and y .The algorithm begins with the identification of k , the number of clusters, using k = (cid:100)|C| /N (cid:101) .The k cluster centers are then initialized randomly using the k -means++ method by Arthur andVassilvitskii (2007). The method first selects a point uniformly at random from C as the first center, u , and then selects the i th center, u i , from C with probability S ( u i ) / [ (cid:80) c ∈C S ( c ) ] until k centersare selected. Each point c ∈ C is then assigned to its nearest cluster center subject to the constraintthat each center is assigned at most N points. This assignment step is accomplished by solving thegeneralized-assignment problem described in Figure 3. The formulation uses a binary variable x c , u that indicates a point c is assigned to center u when set. Its objective function (9) minimizes thetotal distance between all points and their assigned centers. Constraints (10) assign each point toa center, while constraints (11) limit the number of points assigned to each center by N .The assignment step is followed by an update step which recalculates the coordinates of eachcluster center by averaging the coordinates of its assigned points: u = (cid:80) c ∈C x c , u c (cid:80) c ∈C x c , u ∀ u ∈ U (13)The assignment and update steps are then repeated until the point-center assignments stabilize,i.e., until the centers every point are assigned to remain the same in consecutive iterations. asan and Van Hentenryck: Commuting with Autonomous Vehicles min (cid:88) c ∈C (cid:88) u ∈U S ( c , u ) x c , u (9)s.t. (cid:88) u ∈U x c , u = 1 ∀ c ∈ C (10) (cid:88) c ∈C x c , u ≤ N ∀ u ∈ U (11) x c , u ∈ { , } ∀ c ∈ C , ∀ u ∈ U (12) Figure 3 The Clustering Formulation.
5. The Commute Trip Sharing Problem for Autonomous Vehicles
This section specifies the CTSPAV, a problem which seeks a set of minimal cost AV routes to serveevery inbound and outbound trip of a set of commuters, C . Let n denote the total number of commuters, i.e., n = |C| . For every commuter i ∈ C , let nodes i , n + i , 2 n + i , and 3 n + i represent the inbound pickup, inbound drop-off, outbound pickup, andoutbound drop-off locations of the rider trips respectively. Then let the sets of all inbound pickup,all inbound drop-off, all outbound pickup, and all outbound drop-off nodes be denoted by P + = { , . . . , n } , D + = { n + 1 , . . . , n } , P − = { n + 1 , . . . , n } , and D − = { n + 1 , . . . , n } respectively.Furthermore, let P = P + ∪ P − and D = D + ∪ D − . With this notation, note that n + i provides thedrop-off node corresponding to any pickup node i ∈ P . By definition of AV routes, the followingprecedence constraints apply to the following set of nodes: i ≺ n + i ≺ n + i ≺ n + i ∀ i ∈ P + (14)where i ≺ j denotes the precedence relation between nodes i and j , i.e., the constraint indicatingthat i must be visited before j if both i and j are served by the same AV route.The directed graph G = ( N , A ) with the node set N = P ∪ D ∪ { v s , v t } contains all pickup anddrop-off nodes together with a source and a sink node (both representing the designated depot)and its edge set A = { ( i, j ) : i, j ∈ N , i (cid:54) = j } consists of all possible edges as a first approximation.A time window [ a i , b i ] and a service duration s i are then associated with each node i ∈ P ∪ D .No time windows are associated with v s and v t as it is assumed that the AVs may start and endtheir routes at any time of the day. Additionally, a ride-duration limit L i is associated with eachnode i ∈ P . Finally, a travel time τ ( i,j ) , a distance ς ( i,j ) , and a cost c ( i,j ) are associated with eachedge ( i, j ) ∈ A , and δ + ( i ) and δ − ( i ) denote the sets of all outgoing and incoming edges of node i respectively. asan and Van Hentenryck: Commuting with Autonomous Vehicles This section introduces a MIP model for the CTSPAV. The MIP is summarized in Figure 4: itformalizes the CTSPAV and is defined on the graph G and the set Ω of all feasible mini routes .The MIP formulation uses two sets of binary variables: variable X r indicates whether mini route r ∈ Ω is selected and variable Y ( i,j ) indicates whether edge ( i, j ) ∈ A is used, i.e., whether node j should be visited immediately after node i by an AV route in the optimal solution. Additionally,the model uses a continuous variable T i that represents the start of service time at node i ∈ P ∪ D .The objective function (17) minimizes the total cost of all selected edges. Contraints (18) ensureeach trip is served by exactly one mini route, while constraints (19) select edges belonging toselected mini routes. Constraints (20) and (21) simultaneously ensure each pickup and drop-offnode is visited exactly once while conserving the flow through each. Constraints (22) and (23)ensure the start of service time at the tail and head of every selected edge is compatible with thetravel time along the edge using large constants M ( i,j ) and ¯ M ( i,j ) . Finally, constraints (24) and (25)describe the ride-duration limit of every trip and the time-window constraint of every pickup anddrop-off node respectively.Note that constraints (22) and (23) are generalizations of the popular Miller-Tucker-Zemlin(MTZ) subtour-elimination constraints for the TSP (Miller et al. 1960). They utilize big- M con-stants and enforce the underlying constraints on a subset of edges: M ( i,j ) = max { , b i + s i + τ ( i,j ) − a j } ∀ i, j ∈ P ∪ D (15)¯ M ( i,j ) = max { , b j − a i − s i − τ ( i,j ) } ∀ i ∈ P ∪ D , ∀ j ∈ D (16)The model adopts a lexicographic objective whose primary objective is to minimize the numberof vehicles used and whose secondary objective is to minimize the total travel distance. Thislexicographic ordering is accomplished by weighting the sub-objectives: an identical, large fixedcost and a variable cost that is proportional to the route total distance are assigned to each AVroute. The edge costs are defined as follows to accomplish this goal: c e = (cid:40) ς e + 100ˆ ς max ∀ e ∈ δ + ( v s ) ς e otherwise (28)where ˆ ς max is a constant equal to the length (total distance) of the longest AV route. Letting R denote the set of all feasible AV routes, ˆ ς max is given by:ˆ ς max = max ρ ∈R (cid:88) ( i,j ) ∈ ρ ς ( i,j ) (29)The CTSPAV model essentially solves a scheduling problem that selects and assembles feasible miniroutes to form longer, feasible AV routes to cover all trips and minimize the total cost. The optimalAV routes are obtained by constructing paths beginning at v s and ending at v t from the selectededges, and the start and end times can be calculated using Equations (6) and (8) respectively. asan and Van Hentenryck: Commuting with Autonomous Vehicles min (cid:88) e ∈A c e Y e (17)subject to (cid:88) r ∈ Ω: i ∈ r X r = 1 ∀ i ∈ P (18) (cid:88) r ∈ Ω: e ∈ r X r − Y e ≤ ∀ e ∈ A \ { δ + ( v s ) ∪ δ − ( v t ) } (19) (cid:88) e ∈ δ + ( i ) Y e = 1 ∀ i ∈ P ∪ D (20) (cid:88) e ∈ δ − ( i ) Y e = 1 ∀ i ∈ P ∪ D (21) T i + s i + τ ( i,j ) ≤ T j + M ( i,j ) (1 − Y ( i,j ) ) ∀ i, j ∈ P ∪ D (22) T i + s i + τ ( i,j ) ≥ T j − ¯ M ( i,j ) (1 − Y ( i,j ) ) ∀ i ∈ P ∪ D , ∀ j ∈ D (23) T i + n − ( T i + s i ) ≤ L i ∀ i ∈ P (24) a i ≤ T i ≤ b i ∀ i ∈ P ∪ D (25) X r ∈ { , } ∀ r ∈ Ω (26) Y e ∈ { , } ∀ e ∈ A (27) Figure 4 The MIP Model for the CTSPAV.
Since the MIP model is defined in terms of all mini-routes, this section describes the Mini Route-Enumeration Algorithm (MREA), a procedure for enumerating all the feasible mini routes in Ωthat is based on the algorithm proposed by Hasan et al. (2020). The set Ω can be partitionedinto Ω = Ω + ∪ Ω − , where Ω + represents the set of feasible inbound mini routes (which coversonly inbound trips) while Ω − represents the set of feasible outbound mini routes (which coversonly outbound trips). Without loss of generality, this section only describes the procedure forenumerating the mini routes in Ω + .The procedure is summarized in Algorithm 1. It requires as inputs the set T + of all inboundtrips and the vehicle capacity K . It begins by considering all feasible inbound mini routes for avehicle capacity of 1 by adding the routes for all direct trips from T + to Ω + (lines 2–3). It thenenumerates feasible routes for progressively increasing vehicle capacities by increasing a parameter k which represents the current vehicle capacity from 2 to K (line 4). For each k , the procedurefirst enumerates all k -combinations of trips from T + (line 5). Let Q k represent the set of all k -tripcombinations. It then enumerates all valid mini routes for each trip combination q ∈ Q k . Let Ω vq be asan and Van Hentenryck: Commuting with Autonomous Vehicles Algorithm 1
Mini Route-Enumeration Algorithm for Ω + Require: T + , K Ω + ← Ø for each t + c ∈ T + do Ω + ← Ω + ∪ { o + c → d + c } for k = 2 to K do Q k ← { all k -combinations of T + } for each q ∈ Q k do Ω vq ← { all valid mini routes of q } for each r + ∈ Ω vq do if f easible ( r + ) then Ω + ← Ω + ∪ { r + } return Ω + this set of routes for a trip combination q . The procedure checks the feasibility of each route in Ω vq (using the f easible function) and adds the ones that are feasible to Ω + (lines 8–10).The labeling procedure by Gschwind and Irnich (2015) makes it possible to check feasibility whenextending partial mini routes and permits a more efficient implementation of lines 7–10. The setof feasible mini routes for any trip combination q can be enumerated by performing a depth-firstsearch which checks the feasibility of each partial route as it is being extended and backtrackswhen an extension is infeasible. Furthermore, the independence of the search procedure for eachtrip combination q ∈ Q k allows each combination to be performed in parallel.In summary, the enumeration procedure considers all trip combinations of size k ≤ K (of whichthere are O ( n K ) combinations). For each k -combination, it enumerates ( k !) valid route permu-tations ( k ! pickup node permutations followed by k ! drop-off node permutations for each pickuppermutation) and checks the feasibility of each. The procedure therefore has a time and spacecomplexity of O ([ K !] n K ). Hasan and Van Hentenryck (in press 2021) have shown that capacitiesgreater than 5 bring only marginal benefits for the case study, which will also be confirmed laterin this paper. G Graph G can be made more compact by only retaining edges that satisfy a priori route-feasibilityconstraints. This is done by pre-processing time-window, pairing, precedence, and ride-durationlimit constraints on A to identify and eliminate edges that are infeasible, i.e., those that cannotbelong to any feasible AV route. In this work, the set of infeasible edges is identified using a asan and Van Hentenryck: Commuting with Autonomous Vehicles combination of rules proposed by Dumas et al. (1991) and Cordeau (2006). These rules are presentedin the Appendix.
6. Valid Inequalities for the CTSPAV
The CTSPAV MIP is solved with a traditional branch-and-cut procedure that expoits a num-ber of valid inequalities for the MIP formulation. The inequalities are valid for all nodes in thebranch and bound tree, and the LP relaxation at each node incorporates all inequalities discov-ered up to that point. Numerous families of valid inequalities, that have been proposed for theTSP (Dantzig et al. 1954, Gr¨otschel and Padberg 1985, Padberg and Rinaldi 1991), ATSP (Fis-chetti and Toth 1997), PCATSP (Balas et al. 1995), PDP (Ruland and Rodin 1997), ATSPTW(Ascheuer et al. 2000, 2001), VRPTW (Kohl et al. 1999, Bard et al. 2002, Kallehauge et al. 2007),PDPTW (Ropke and Cordeau 2009), and DARP (Cordeau 2006), are also valid for the CTSPAVas the CTSPAV is a generalization of the DARP. However, this work only considers inequalitiesthat specifically improve the lower bound on the vehicle count (the primary objective). This isbecause extensive computational experiments from an earlier work (Hasan and Van Hentenryckin press 2021) showed that the LP relaxation already provides a sufficiently strong lower boundfor the secondary objective (total distance). This section describes the considered valid inequal-ities with their respective separation heuristics when applicable. The following notation is usedto simplify the exposition. For any set of edges A (cid:48) ⊆ A , let Y ( A (cid:48) ) = (cid:80) e ∈A (cid:48) Y e . For a set of nodes S ⊆ N , let ¯ S denote its complement, i.e., ¯ S = { i ∈ N | i / ∈ S } . For any two node sets S, T ⊆ N , let(
S, T ) = { ( i, j ) ∈ A | i ∈ S, j ∈ T } . For brevity, Y ( S, T ) is used to represent Y (( S, T )). Finally, fornode set S ⊆ P ∪ D , let π ( S ) = { i ∈ P | n + i ∈ S } and σ ( S ) = { n + i ∈ D | i ∈ S } denote the sets ofpredecessors and successors of S respectively. Suppose that a (fractional) lower bound χ LB is known for the vehicle count. The inequality Y ( δ + ( v s )) ≥ (cid:100) χ LB (cid:101) (30)is a direct consequence of the integrality of the vehicle count. Such a lower bound can be obtainedby selecting the best bound in the branch-and-bound algorithm. Let Y ∗ e denote the value of Y e inthe LP-relaxation for this best bound. The lower bound χ BB can be obtained by χ BB = (cid:88) e ∈ δ + ( v s ) Y ∗ e (31)and used in place of χ LB in (30). asan and Van Hentenryck: Commuting with Autonomous Vehicles A stronger lower bound may be obtained from a column-generation procedure that solves theCTSPAV as a DARP. This recognition is based on an earlier work (Hasan and Van Hentenryckin press 2021) which discovered that a column-generation procedure which resembles that used byGschwind and Irnich (2015) for solving the DARP is capable of producing strong lower bounds forthe vehicle count of the CTSPAV when it is paired with an appropriate objective function. Thiswork leverages the procedure to strengthen the vehicle-count lower bound of the CTSPAV MIP.The DARP column-generation procedure of Hasan and Van Hentenryck (in press 2021) featuresa Pricing Subproblem (PSP) that searches for AV routes with negative reduced costs to improvethe objective function of a set-covering master problem (MP) whose columns consist of the routes.More specifically, it utilizes a restricted master problem (RMP) which is the linear relaxation ofthe MP that is defined on a subset R (cid:48) ⊆ R of all feasible AV routes. The discovered routes areprogressively added to R (cid:48) as the RMP and the PSP are solved iteratively. The column generationterminates when the PSP cannot produce AV routes with negative reduced costs. At this stage, theobjective value z RMP of the RMP is identical to the optimal objective z ∗ of the linear relaxation ofthe original MP. In this work, the column-generation procedure is not used to obtain a solution tothe CTSPAV per se; instead it is used to extract (potentially strong) lower bounds to the primaryobjective of the CTSPAV. The following describes the procedure for obtaining these lower bounds. The Restricted Master Problem
The RMP is a set-covering formulation:min z = (cid:88) ρ ∈R (cid:48) X ρ (32)subject to (cid:88) ρ ∈R (cid:48) a i,ρ X ρ ≥ ∀ i ∈ P (33) X ρ ≥ ∀ ρ ∈ R (cid:48) (34)It is defined on a subset R (cid:48) ⊆ R of all feasible AV routes, and uses a variable X ρ to indicatewhether AV route ρ ∈ R (cid:48) is used in the optimal solution. Its objective function (32) minimizes thenumber z of selected AV routes and is therefore identical to the primary objective of the CTSPAV.Constraints (33) ensure each pickup node is covered in the solution, and a i,ρ is a constant thatindicates the number of times node i is visited by route ρ . The Pricing Subproblem
The PSP searches for AV routes with negative reduced costs to be addedto R (cid:48) . It uses { µ i : i ∈ P} , the set of optimal duals of constraints (33), to compute the reducedcosts of the undiscovered routes. The reduced cost of a route ρ is given by¯ c ρ = 1 − (cid:88) i ∈P a i,ρ µ i . (35) asan and Van Hentenryck: Commuting with Autonomous Vehicles To find these routes, a graph G identical to that defined in Section 5 is first constructed. A reducedcost ¯ c ( i,j ) is then associated with each edge ( i, j ) ∈ A , and it is defined as follows so that the totalcost of any path in G from v s to v t is equivalent to (35):¯ c ( i,j ) = ∀ ( i, j ) ∈ δ + ( v s ) − µ i ∀ i ∈ P , ∀ j ∈ N ∀ i ∈ D , ∀ j ∈ N . (36)Obtaining a solution to the PSP is then a matter of finding a feasible AV route, i.e., a path from v s to v t that satisfies the time-window, capacity, pairing, precedence, and ride-duration limit constraints,with negative reduced cost. The PSP can be solved by first finding the least-cost feasible pathfrom v s to v t and then adding it to R (cid:48) if the cost is negative. This approach makes the probleman ESPPRC which can be solved by the label-setting dynamic program proposed by Gschwindand Irnich (2015). The necessity of ensuring elementarity of the path (to ensure its feasibility),however, makes the problem especially hard to solve (Dror 1994). Since we are only interested inderiving lower bounds to the vehicle count from this procedure and not in discovering AV routesper se, the elementarity requirement can be relaxed to admit a pseudo-polynomial solution from thelabel-setting algorithm. While the relaxation, in theory, may cause z RMP to converge to a weakerprimal bound as the PSP admits a larger set of routes R (cid:48)(cid:48) ⊇ R (cid:48) , other works that have adopted asimilar strategy (e.g., Ropke and Cordeau (2009) and Gschwind and Irnich (2015)) have discoveredthat the lower bound is only slightly weaker in practice. Extracting a Lower Bound to the Vehicle Count from the PSP
As mentioned earlier, z RMP con-verges to z ∗ and therefore becomes a valid lower bound to the vehicle count of the CTSPAV whenthe PSP is unable to discover a new AV route with negative reduced cost. However, reaching thispoint in the procedure typically requires many column-generation iterations and thus a long com-putation time. Prior to it, z RMP only represents an upper bound to z ∗ and therefore it cannot beused to bound the vehicle count. Fortunately, the identical unit cost of each AV route in the RMPallows for the derivation of a lower bound to z ∗ using the method proposed by Farley (1990). TheFarley bound after the k th column-generation iteration is given by: z k Farley = z RMP − ¯ c kρ (37)where ¯ c kρ represents the smallest route reduced cost discovered by the PSP after the k th iteration.As the value of z k Farley tends to fluctuate between iterations, a monotonically non-decreasing lowerbound to z ∗ can be obtained with the following equation: z k Farley = max (cid:26) z RMP − ¯ c kρ , z k − (cid:27) (38) asan and Van Hentenryck: Commuting with Autonomous Vehicles As z k Farley is a lower bound to z ∗ , it is also a valid lower bound to the vehicle count of theCTSPAV. Therefore, χ LB for cut (30) may be defined as follows: χ LB = max (cid:8) χ BB , z k Farley (cid:9) (39)Since z k Farley as defined in (38) is monotonically non-decreasing and improves with the number ofcolumn-generation iterations, it is practical to dedicate a single thread for executing this column-generation procedure and use the remaining thread(s) for solving the CTSPAV MIP in parallel. TheCTSPAV MIP may then check for the most up-to-date value of z k Farley from the column-generationthread after evaluating the LP relaxation of each tree node and introduce cut (30) when there isan improvement to the rounded lower bound.
The two-path inequality was originally conceived by Kohl et al. (1999) for the VRPTW. It hasbeen shown to be particularly effective at strengthening the lower bound for the vehicle count ofthe VRPTW (Bard et al. 2002) and the PDPTW (Ropke and Cordeau 2009) when vehicle-countminimization is (part of) the objective function. For a set of nodes S ⊆ P ∪ D , let κ ( S ) denotethe minimum number of vehicles needed to serve S , i.e., the minimum number of vehicles neededto serve all nodes in S while satisfying all route-feasibility constraints. The following two-pathinequality, Y ( S, ¯ S ) ≥ ∀ S ⊆ P ∪ D , κ ( S ) > S , i.e., when κ ( S ) > Y ( S, ¯ S ) ≥ ∀ S ⊆ P ∪ D , | S | ≥ Y ( S, S ) ≤ | S | − ∀ S ⊆ P ∪ D , | S | ≥ S with two or more nodes. The two-path inequality (40) can therefore be seen as a strengthenedSEC as it requires at least two units of flow emanating from any set S , however its validity alsorequires a stronger condition, i.e., κ ( S ) > asan and Van Hentenryck: Commuting with Autonomous Vehicles Figure 5 Graph G S (Each Dotted Line Represents a Pair of Bidirectional Edges). For the CTSPAV, a method similar to that proposed in Ropke and Cordeau (2009) may be usedto determine if κ ( S ) > S . It essentially requires one to determine if there existsa feasible path that first visits all the nodes in π ( S ) \ S , followed by all the nodes in S , and thenall the nodes in σ ( S ) \ S . If the path does not exist, then κ ( S ) >
1. The task of determining theexistence of this path can be accomplished by first constructing a three-layered graph G S = ( N S , A S )with nodes N S = π ( S ) ∪ S ∪ σ ( S ) ∪ { v s , v t } and an initially empty edge set A S . The nodes from N S \ { v s , v t } are grouped into three layers, the first consisting of π ( S ) \ S , the second consisting of S , and the third containing σ ( S ) \ S . The following sets of edges are then introduced into A S : • { ( v s , v t ) } • ( { v s } , π ( S ) \ S ) ∩ A • ( π ( S ) \ S, S ) ∩ A • ( S, σ ( S ) \ S ) ∩ A • ( σ ( S ) \ S, { v t } ) ∩ A • ( π ( S ) \ S, π ( S ) \ S ) ∩ A • ( S, S ) ∩ A • ( σ ( S ) \ S, σ ( S ) \ S ) ∩ A where A denotes the set of feasible edges of graph G (after they have been filtered). Figure 5provides a sketch of G S . The following sets of edges are introduced into A S should either π ( S ) \ S or σ ( S ) \ S be empty: • If π ( S ) \ S = Ø, introduce ( { v s } , S ) ∩ A • If σ ( S ) \ S = Ø, introduce ( S, { v t } ) ∩ A One now needs to determine if there exists a feasible path from v s to v t that visits every node of G S . This problem can be treated as an ESPPRC, whereby an edge cost of − N S (i.e., edges in ( π ( S ) , N S \ π ( S ))). A feasible path from v s asan and Van Hentenryck: Commuting with Autonomous Vehicles to v t that visits every node of G S then exists if and only if the least-cost elementary path from v s to v t has a total cost of −| π ( S ) | . While this ESPPRC is well-known to be NP-hard (Dror 1994),it can be solved efficiently using the label-setting algorithm by Gschwind and Irnich (2015) forsmall S . Therefore, one just needs to solve the ESPPRC and check the total cost of the resultingelementary path. Should it be greater than −| π ( S ) | , then the nodes of S cannot be feasibly servedby a single vehicle, κ ( S ) >
1, and (40) becomes a valid inequality.
The separation heuristic for the two-path inequalities first iden-tifies sets of nodes S for which κ ( S ) >
1. As the two-path inequality is essentially a strengthenedSEC, the heuristic utilized in this work first identifies sets of nodes that form subtours (cycles) inthe LP-relaxation solution at each tree node. Let Y ∗ e denote the value of Y e from the solution ofthe LP relaxation. For every subtour S considered, the heuristic then checks if (cid:80) e ∈ ( S, ¯ S ) Y ∗ e < κ ( S ) >
1. Satisfaction of these two conditions indicates that the two-path inequality is validfor S , and that it is violated by S in the LP-relaxation solution. The heuristic therefore adds thetwo-path inequality to eliminate generation of the subtour from subsequent LP solutions.To identify subtours from the LP relaxation at each tree node, the heuristic by Drexl (2013) isused. The heuristic was proposed as a cheaper yet effective alternative for identifying violated SECsto the exact method proposed by Gomory and Hu (1961), as it has an O ( n ) complexity comparedto the O ( n ) complexity of the latter. For any LP solution, a support graph, G sp = ( N sp , A sp ), isfirst constructed with nodes N sp = N and edges A sp = { e ∈ A | Y ∗ e > } . All strongly-connectedcomponents (SCCs) of G sp are then identified, where an SCC of a graph is its subgraph with morethan one node whereby there exists a path between all pairs of its nodes. The rationale behindidentification of SCCs is that each forms a subtour (the nodes of the SCC form a cycle(s) as everynode is reachable from another). In practice, all SCCs of G sp can be computed using the algorithmby Tarjan (1972) which has a time complexity of O ( |N sp | + |A sp | ). Let S sp denote the set of allSCCs of G sp , and for each SCC c ∈ S sp , let S c denote its set of nodes. For every c ∈ S sp , the heuristicthen checks if the total flow leaving S c is less than 2, i.e., if (cid:80) e ∈ ( S c , ¯ S c ) Y ∗ e <
2. If this condition issatisfied for S c , the heuristic then determines if κ ( S c ) > Y ( S c , ¯ S c ) ≥ κ ( S c ) > κ ( S c ) >
1, results of the procedurefor every set S c are stored in a hash table, and the hash table is examined first before the procedureis performed on any set S to ensure that the same calculations are not repeated. Furthermore, thepart of the procedure which solves an ESPPRC on graph G S can also be made more efficient. Insteadof directly applying the label-setting algorithm of Gschwind and Irnich (2015) which proposeskeeping track of all visited pickup nodes and preventing path extensions to the already visited nodes asan and Van Hentenryck: Commuting with Autonomous Vehicles to ensure elementarity, the procedure proposed by Boland et al. (2006) can be used. The latterentails iteratively solving a sequence of relaxed SPPRCs, whereby the elementarity requirementis completely relaxed in the very beginning. A repeated node from the solution of the relaxedproblem is selected and added to a set U , after which the problem is solved again, this time with anadditional restriction that the nodes in U can only be visited once. The procedure is repeated with U being progressively enlarged until an elementary path is discovered. The rationale behind thisprocedure is that solving the sequence of relaxed SPPRCs is usually less expensive than solving asingle ESPPRC in practice, as often times the former discovers an elementary path without havingto include all pickup nodes in the set U . Desaulniers et al. (2008) proposed adding only the firstrepeated node from the solution of the relaxed problem to U after each iteration, and our initialevaluations show that this approach works very well in practice. Predecessor and successor inequalities were first introduced by Balas et al. (1995) for the PCATSP.The predecessor inequality ( π -inequality) is given by: Y ( S \ π ( S ) , ¯ S \ π ( S )) ≥ ∀ S ⊆ P ∪ D , | S | ≥ σ -inequality) is given by: Y ( ¯ S \ σ ( S ) , S \ σ ( S )) ≥ ∀ S ⊆ P ∪ D , | S | ≥ The heuristic utilized to separate π - and σ -inequalities is verysimilar to that described in Section 6.3.1 for the two-path inequality. At each tree node, values of Y ∗ e are first used to construct a support graph G sp , after which S sp which represents the set of all SCCsof G sp are identified. For each c ∈ S sp , the heuristic then checks if either inequalities (43) or (44)have been violated for S c , i.e., if either Y ( S c \ π ( S c ) , ¯ S c \ π ( S c )) < Y ( ¯ S c \ σ ( S c ) , S c \ σ ( S c )) < π - or σ -inequalities are introduced to the MIP for each violation. The lifted MTZ inequality was initially proposed by Desrochers and Laporte (1991) for theVRPTW. They were intended to strengthen MTZ constraints that are similar to (22) and (23)which are well-known to produce weak LP relaxations (Langevin et al. 1990, Gouveia and Pires1999). The MTZ constraints for an edge ( i, j ) is strengthened by taking into consideration the flowalong the opposite edge ( j, i ) combined with the fact that only one of the edges may have positive asan and Van Hentenryck:
Commuting with Autonomous Vehicles flow in a feasible integer solution. The lifted versions constraints (22) and (23) are given by (45)and (46) respectively. T i + s i + τ ( i,j ) ≤ T j + M ( i,j ) (1 − Y ( i,j ) ) − α ( j,i ) Y ( j,i ) ∀ i, j ∈ P ∪ D (45) T i + s i + τ ( i,j ) ≥ T j − ¯ M ( i,j ) (1 − Y ( i,j ) ) − β ( j,i ) Y ( j,i ) ∀ i ∈ P ∪ D , ∀ j ∈ D (46)To correctly lift the constraints using this technique, the coefficients of the flow variable of theopposite edge, α ( j,i ) and β ( j,i ) , are assigned values that are as large as possible while ensuring thatinequalities (45) and (46) are still valid for any feasible integer solution. Desrochers and Laporte(1991) proposed coefficient values for the VRPTW that ensure the earliest start of service timesfor every node. As serving pickup nodes as early as possible may not be desirable for the CTSPAV(as doing so lengthens the ride duration of the picked-up rider and thus increases the likelihood ofexceeding her ride-duration limit), the coefficients are adjusted to (47) and (48) for the CTSPAV. α ( j,i ) = (cid:40) M ( i,j ) − s i − τ ( i,j ) − s j − τ ( j,i ) if i ∈ D M ( i,j ) − s i − τ ( i,j ) − b i + a j otherwise (47) β ( j,i ) = − ¯ M ( i,j ) − s i − τ ( i,j ) − s j − τ ( j,i ) (48)The validity of the lifted constraints can be verified by first substituting (47) and (48) into (45) and(46) respectively, and then setting the flows along edges ( i, j ) and ( j, i ) to zero or setting the flowalong either edge to one. Firstly, setting both Y ( i,j ) and Y ( j,i ) to zero just disables constraints (45)and (46) for both edges. Next, setting Y ( i,j ) = 1 and Y ( j,i ) = 0 produces the following constraints, T i + s i + τ ( i,j ) ≤ T j if i, j ∈ P ∪ D (49) T i + s i + τ ( i,j ) ≥ T j if i ∈ P ∪ D , j ∈ D (50)which simply enforce the increasing service time requirement along edge ( i, j ). Finally, setting Y ( i,j ) = 0 and Y ( j,i ) = 1 results in the following set of constraints: T j + s j + τ ( j,i ) ≥ T i if j ∈ P ∪ D , i ∈ D (51) T i − T j ≤ b i − a j if j ∈ P ∪ D , i ∈ P (52) T j + s j + τ ( j,i ) ≤ T i if j ∈ D , i ∈ P ∪ D (53)Constraints (51) and (53) simply enforce increasing service times along edge ( j, i ), while (52) isobviously a valid inequality if edge ( j, i ) is selected. asan and Van Hentenryck: Commuting with Autonomous Vehicles The lifted time-bound inequalities were also proposed by Desrochers and Laporte (1991) tostrengthen the time-window constraints of the VRPTW. Inequalities (54) and (55) strengthen thetime-window constraints of node i by taking into consideration the temporal requirements alongthe node’s incoming and outgoing edges with positive flow. T i ≥ a i + (cid:88) ( j,i ) ∈ δ − ( i ) max { , a j − a i + s j + τ ( j,i ) } Y ( j,i ) ∀ i ∈ P ∪ D (54) T i ≤ b i − (cid:88) ( i,j ) ∈ δ + ( i ) max { , b i − b j + s i + τ ( i,j ) } Y ( i,j ) ∀ i ∈ P ∪ D (55)
7. Computational Results
This section presents the computational results of the branch-and-cut algorithm on probleminstances derived from a real-world dataset of commute trips.
Three variants of the branch-and-cut algorithm are considered and contrasted in the evaluations;they are named CTSPAV
Base , CTSPAV
SEC , and CTSPAV
Hybrid . Each is differentiated by the typesof valid inequalities included in its implementation. They are specificied as follows: • CTSPAV
Base is the core algorithm and implements the simplest valid inequalities: lifted timebounds, lifted MTZ, and rounded vehicle count which uses χ BB as its lower bound; • CTSPAV
SEC is CTSPAV
Base with the two-path, predecessor, and successor inequalities; • CTSPAV
Hybrid is CTSPAV
Base with the DARP lower bound from Section 6.1.The latter variant also uses the interior-point, dual-stabilization method proposed by Rousseauet al. (2007) to accelerate the convergence of its column-generation procedure. Furthermore, insteadof only selecting the least-cost feasible path with negative reduced cost in its PSP, all non-dominatedpaths resulting from the label-setting algorithm with negative reduced costs are added to R (cid:48) tofurther accelerate convergence. Problem instances for the computational evaluations are derived from the commute trip datasetfirst used by Hasan et al. (2018). It consists of the real-world arrival and departure times to 15parking structures located in downtown Ann Arbor, Michigan, of approximately 15,000 commutersthat were collected throughout the month of April 2017. This information, when joined with thehome addresses of every commuter, allowed the reconstruction of their daily commute trips. Theperformance evaluations utilize the trips made by commuters living within Ann Arbor’s city limits,the region bounded by highways US-23, M-14, and I-94. More specifically, the 2,200 commute tripsfrom this region made on the busiest day of the month (Wednesday of week 2) were first selected asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 1 Parameters for Constructing Problem Instances
Problem size N ∆ R K
Number of instancesLarge 100 10 mins 0.50 4 22Medium 75 10 mins 0.50 4 30Tight 100 5 mins 0.25 4 22and then partitioned into smaller problem instances using the clustering algorithm described inSection 4. Trip sharing is then only considered intra-cluster with the largest parking structurearbitrarily designated as the depot for all clusters.In addition to this, the following assumptions are made in order to define the time windowsand ride-duration limits of each trip. Consistent with past works on the DARP (e.g., Jaw et al.(1986), Cordeau and Laporte (2003b), Cordeau (2006)), each rider i specifies a desired arrival time at + i at the destination of her inbound trip and a desired departure time dt − i at the origin of heroutbound trip when requesting a trip. Riders also tolerate a maximum shift of ± ∆ to the desiredtimes. By considering the arrival and departure times to and from the parking structures as thedesired times, an arrival-time upper bound at node n + i of b n + i = at + i + ∆ and a time window atnode 2 n + i of [ a n + i , b n + i ] = [ dt − i − ∆ , dt − i + ∆] are defined for each i ∈ P + . Consequently, the timewindow at node i is given by [ a i , b i ] = [ b n + i − s i − L i − , b n + i − s i − L i ] and the arrival-time upperbound at node 3 n + i is given by b n + i = b n + i + s n + i + L n + i for each i ∈ P + . Finally, consistentwith Hunsaker and Savelsbergh (2002), the ride-duration limit of each trip is defined as an R %extension to the direct trip, i.e., L i = (1 + R ) τ i,n + i for each i ∈ P .A set of tight, medium, and large problem instances are constructed by varying parameter N in the clustering algorithm together with ∆ and R . The parameter combinations are carefullyselected to highlight performance differences in the three variants of the branch-and-cut algorithmconsidered. A vehicle capacity of K = 4 is used in all instances to represent the use of autonomouscars. Table 1 shows the parameters used together with the number of instances created when theclustering algorithm is applied on the set of 2,200 commuters: All algorithms are implemented in C++. Parallelization of the mini route-enumeration algorithmis handled with OpenMP, while the parallel execution of the column-generation procedure and theMIP of CTSPAV
Hybrid is handled with the thread class from the C++11 standard library. All LPsand MIPs are solved with Gurobi 9.0.2, while graph algorithms from the Boost Graph Library(version 1.70.0) are used to calculate SCCs of a graph and to implement the label-setting algorithmof Gschwind and Irnich (2015). Gurobi’s callback feature is used to implement the bespoke cutting-plane separation and insertion, while the MIP solver is configured with its default parameters. asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 2 Average Vehicle Count and Optimality Gaps of Every CTSPAV Variant for Every Problem Size
CTSPAVvariant Average vehicle count gap Average optimality gapLarge Medium Tight Large Medium TightHybrid 1.18 0.50 0.00 31.8% 16.6% 0.0%SEC 1.73 0.73 0.09 45.5% 23.8% 1.7%Base 2.50 1.67 0.14 68.0% 59.0% 3.2%For problem instance construction, Geocodio is used to geocode GPS coordinates of every addressconsidered, after which GraphHopper’s Directions API is used in conjunction with OpenStreetMapdata to estimate the shortest path, travel time, and travel distance between any two nodes. Unlessstated otherwise, every problem instance is solved on a compute cluster, each utilizing 4 cores of a3.0 GHz Intel Xeon Gold 6154 processor and 16 GB of RAM. All four cores are used for the MREA.For CTSPAV
Hybrid , one core is dedicated for the column-generation procedure while the remainingthree are used for solving the MIP. All four cores are used for solving the MIPs of CTSPAV
SEC and CTSPAV
Base . Finally, a 2-hour time budget is allocated for solving all MIPs.
Table 2 first summarizes the average vehicle count gaps and average optimality gaps obtained forevery problem size and every CTSPAV variant. χ MIP , z MIP , and z BB denote the vehicle count, theobjective value of the best incumbent solution, and its best bound respectively. The vehicle countgap is given by χ MIP − (cid:100) χ LB (cid:101) , while the optimality gap is given by ( z MIP − z BB ) /z MIP . The completeresults of all the computational experiments are listed in Tables 3–8 in the Appendix. Note that theroute enumeration times for every problem instance are consistently less than 60 seconds, whichhighlights the efficiency of the MREA.The average optimality gaps for large and medium instances appear to be relatively large. How-ever, a closer examination paints a different picture, as their values are relatively small across theboard. In fact, the average count gap for CTSPAV
Hybrid is only a little above one for the largeproblem instances, and is less than one for the tighter instances. The values for CTSPAV
Hybrid are also consistently smaller across the board than those of CTSPAV
SEC which, in turn, aresmaller than those of CTSPAV
Base . This observation provides the first evidence of the capabilityof CTSPAV
Hybrid ’s column-generation procedure at producing very strong lower bounds for theprimary objective; it also demonstrates the effectiveness of the combination of the two-path, succes-sor, and predecessor inequalities at closing the vehicle count gap (compared to an implementationthat only adopts the three basic inequalities). While the latter set of inequalities produces signifi-cant improvements in closing the primary gap, they are nevertheless outperformed by the roundedvehicle-count inequalities of CTSPAV
Hybrid . asan and Van Hentenryck: Commuting with Autonomous Vehicles I n s t a n ce c o un t Problem size
Hybrid SEC Base
Figure 6 Number of Problem Instances Whereby Vehicle Count Gap is Closed by Every CTSPAV Variant.
Figure 6 provides a different perspective by summarizing the number of problem instances whosevehicle count gaps are successfully closed within the 2-hour time limit for every CTSPAV variant.It also displays each count as a fraction of the total number of instances considered. For the largeinstances, CTSPAV
Hybrid could only close the gap for three instances, while the other two variantscould not for any of the problems from the set. This number improves for the medium probleminstances, where CTSPAV
Hybrid could now close the gap for 15 out of the 30 instances, whileCTSPAV
SEC could do the same for 9 of the instances. However, CTSPAV
Base still cannot close theprimary gap for any. Finally, for the tight problem instances, CTSPAV
Hybrid produces the optimalsolution for all of them, while CTSPAV
SEC closes the primary gap for 90.9% of the instances andCTSPAV
Base does the same for 86.4% of them. Regardless of the set of problem instances beingconsidered, the trend is clear: (1) The additional set of inequalities adopted by CTSPAV
SEC allowsit to successfully close the primary gap of more instances than CTSPAV
Base , and (2) CTSPAV
Hybrid consistently outperforms the other two CTSPAV variants at closing the optimality gap. The latterobservation provides yet another evidence of the efficacy of the CTSPAV
Hybrid ’s column-generationprocedure at generating strong lower bounds for the primary objective.Instead of aggregating the results from each problem set, Figures 7, 8, and 9 provide a closer lookat the primary objective value and its corresponding lower bound for every problem instance fromthe large, medium, and tight sets respectively. For instance, Figure 7 shows the best incumbentsolution and the lower bound for the vehicle count of every CTSPAV variant for every largeproblem instance. The figure reveals that, except for a few instances, all three variants producedidentical final vehicle counts. The difference, however, lies in their lower bounds. The lower boundsof CTSPAV
Hybrid dominate those of CTSPAV
SEC in every instance. In turn, those of the latterdominate the lower bounds of CTSPAV
Base in every instance as well. The same observation iscarried over to Figure 8 which summarizes the primary gap of every instance from the mediumset. While CTSPAV
Hybrid and CTSPAV
SEC produce identical lower bounds for more instances from asan and Van Hentenryck:
Commuting with Autonomous Vehicles L L L L L L L L L L L L L L L L L L L L L L V e h i c l e c o un t Instance name
Hybrid best incumbentHybrid lower bound
SEC best incumbentSEC lower bound
Base best incumbentBase lower bound
Figure 7 Best Incumbent Solution and Lower Bound for Vehicle Count of Every CTSPAV Variant for Every LargeProblem Instance. M M M M M M M M M M M M M M M V e h i c l e c o un t Instance name
Hybrid best incumbentHybrid lower boundSEC best incumbentSEC lower boundBase best incumbentBase lower bound
Figure 8 Best Incumbent Solution and Lower Bound for Vehicle Count of Every CTSPAV Variant for EveryMedium Problem Instance. this set, on the whole, lower bounds of CTSPAV
SEC are still dominated by those of CTSPAV
Hybrid .Similarly, they both dominate the lower bounds of CTSPAV
Base . Finally, Figure 9 summarizes theresults of the tight instances, and confirms the observations from the previous two figures. Theobservations from Figures 7, 8, and 9 lead to the following conclusion: Regardless of the size of theproblem considered, there is a clear delineation between the strengths of the lower bounds for theprimary objective of the three CTSPAV variants. CTSPAV
Hybrid dominates CTSPAV
SEC which,in turn, dominates CTSPAV
Base . The relative strength of CTSPAV
Hybrid ’s lower bound directlycontributes to its ability to close or narrow the optimality gap of more problem instances than theother two variants.
Figure 10 presents a closer examination of the evolution of the best bound and best incumbentobjective value of every CTSPAV variant over time for a specific problem instance (instance L0).It also shows the progression of z k Farley (after it has been scaled by 100ˆ ς max ) over time; the lower asan and Van Hentenryck: Commuting with Autonomous Vehicles S S S S S S S S S S S S S S S S S S S S S S V e h i c l e c o un t Instance name
Hybrid best incumbentHybrid lower boundSEC best incumbentSEC lower boundBase best incumbentBase lower bound
Figure 9 Best Incumbent Solution and Lower Bound for Vehicle Count of Every CTSPAV Variant for Every TightProblem Instance. bound is obtained by rounding it to the smallest multiple of 100ˆ ς max . Since the MIP solver, using itsdefault heuristics, is able to discover strong integer solutions fairly quickly for this formulation, thecritical challenge lies in closing the optimality gap quickly. Unfortunately, the CTSPAV formulationuses big- M constants in constraints (22) and (23) which produce weak LP relaxations.The lifted MTZ and lifted time-bound inequalities only provide marginal improvements to theLP relaxation. While the rounded vehicle-count inequality has the capability of rectifying theissue, χ BB rarely becomes fractional in practice, and thus the version of the inequality that onlyuses χ BB as its lower bound rarely improves the vehicle-count lower bound. This explains whyCTSPAV Base always produces the weakest lower bounds. Separation heuristics of the two-path,successor, and predecessor inequalities attempt to alleviate this situation by first searching forsubtours that result from the flow of an LP-relaxation solution, and then introducing the respectiveinequalities to remove these subtour flows from subsequent LP relaxations. The experimental resultsof CTSPAV
SEC demonstrate that these inequalities are indeed effective at further strengtheningthe LP bounds, however the results also show that their effect on the best bound tends to stagnateover time.The CTSPAV
Hybrid attempts to circumvent the CTSPAV formulation’s weak LP bound by dedi-cating a computational thread to solving the same problem using a DARP formulation that focusesonly on the primary objective. The Farley bound z k Farley of the DARP relaxation provides a lowerbound, and its scaled values in Figure 10 show that it progressively improves over time even afterthe best bounds of CTSPAV
Base and CTSPAV
SEC begin to stagnate. The ability of the column-generation to produce relatively stronger lower bounds can be attributed to a few factors:1. The RMP formulation does not utilize any big- M constants.2. The RMP uses only one set of binary variables ( X ρ ), as opposed to two by the CTSPAVMIP ( X r and Y e ). Therefore, fewer convex combinations of its routes are allowed in its LPrelaxation, which leads to stronger primal (and dual) lower bounds. asan and Van Hentenryck: Commuting with Autonomous Vehicles T o t a l c o s t Time (s)
Hybrid best incumbent obj. val.Hybrid best boundSEC best incumbent obj. val.SEC best boundBase best incumbent obj. val.
Base best bound
Ƹ𝜍max 𝑧Farley 𝑘 Figure 10 Evolution of Best Incumbent Objective Value and Best Bound of Every CTSP Variant for ProblemInstance L0
3. Ropke and Cordeau (2006) showed that the set-covering formulation actually implies severalvalid inequalities (precedence and strengthened precedence inequalities) that would otherwiseneed to be enforced explicitly in an edge flow formulation.The approach of dedicating a single thread for executing the column-generation procedure alsohas a side benefit: it allows the branch-and-bound algorithm to freely explore more tree nodeswithout being encumbered by expensive separation heuristics. This is evident from a comparisonof the number of explored nodes for several problem instances, for example, those of CTSPAV
Hybrid and CTSPAV
SEC for instances L1, L5, and L12 from Tables 3 and 4. The results show that theformer was able to explore significantly more nodes, and this could, in turn, lead to the discovery ofbetter integer solutions. While CTSPAV
Hybrid had one fewer thread for solving its MIP, it also didnot have to execute any of the expensive separation heuristics of CTSPAV
SEC which consequentlyresulted in a net gain in terms of the number of nodes it could explore.
It is useful to contrast these results with the column-generation heuristic proposed by Hasan andVan Hentenryck (in press 2021). The heuristic does not exhaustively enumerate all the mini routes asan and Van Hentenryck:
Commuting with Autonomous Vehicles in Ω. Instead it uses a column-generation procedure consisting of a restricted master problem(RMP CTSPAV )—the linear relaxation of MIP model (17)–(27) defined on only a subset Ω (cid:48) ⊆ Ω of themini routes— and a pricing subproblem (PSP
CTSPAV ) that searches for mini routes with negativereduced costs to augment Ω (cid:48) . The RMP
CTSPAV and PSP
CTSPAV are solved repeatedly until thePSP
CTSPAV is unable to find any mini route with negative reduced cost. Then the heuristic solvesthe RMP
CTSPAV as a MIP (that does not incorporate the valid inequalities considered in this work)to obtain a feasible integer solution. Since the heuristic only considers a subset of the feasible miniroutes, it is incapable of proving the optimality of its solution unless the solution of its RMP
CTSPAV at convergence is integral (which is never the case for the instances considered). Nevertheless, it isstill instructive to compare its results against those of the exact CTSPAV
Hybrid method to gaugethe effectiveness of its column-generation procedure at identifying useful mini routes.Tables 9, 10, and 11 (in the Appendix) give comprehensive results for the heuristic on every large,medium, and tight instance respectively. The results show that significantly fewer columns (miniroutes) are considered by the heuristic. On average, it considers 66%, 62%, and 16% fewer columnsfor the large, medium, and tight instances respectively compared to CTSPAV
Hybrid . However, thefinal vehicle counts and total distances of the heuristics and CTSPAV
Hybrid are very similar. Infact, the vehicle count results of the heuristic are identical to those of CTSPAV
Hybrid in all exceptthree instances: L19, M15, and S7. For these three instances, the counts of the heuristic are onlygreater than those of CTSPAV
Hybrid by one vehicle. Moreover, the percentage difference in thetotal distance results are consistently less than 1.50% (on average, they differ by 0.01%). Thissimilarity bodes very well for the heuristic; it highlights the effectiveness of its negative reducedcost criterion for identifying the subset of mini routes that are critical for producing strong integersolutions. It also indicates that the heuristic is more than sufficient for producing high-qualitysolutions, especially in applications whereby proving the optimality of the final solution is notof paramount importance. As mentioned earlier, the heuristic is incapable of closing the vehiclecount or optimality gap for any of the instances, so CTSPAV
Hybrid remains the better candidate inapplications where closing or narrowing the optimality gap is critical.
8. Case Study of Shared Commuting in Ann Arbor, Michigan
This section summarizes the results of a case study that applies the CTSPAV to optimize thecommuting trips from the Ann Arbor dataset. More specifically, it considers all trips (of commutersliving inside and outside city limits) for the first four weekdays (Monday–Thursday) of the busiestweek of April 2017 (week 2). The parameters N , ∆, and R are set to 100, 10 minutes, and 50%respectively for this case study. Its goal is to demonstrate the effectiveness of the CTSPAV at asan and Van Hentenryck:
Commuting with Autonomous Vehicles
12 13 14 15
17 18 19
21 22 23 24 N u m b er o f t r i p s Hour of day
Inbound Outbound (a) Monday
10 11 12
13 14
15 16
17 18
19 20 21
22 23 N u m b er o f t r i p s Hour of day
Inbound Outbound (b) Tuesday
10 11 12
13 14
15 16
17 18
19 20 21
22 23 N u m b er o f t r i p s Hour of day
Inbound Outbound (c) Wednesday
11 12
13 14
15 16
17 18 19
20 21
22 23 N u m b er o f t r i p s Hour of day
Inbound Outbound (d) Thursday
Figure 11 Commute Trip Demand Over 15-Minute Intervals on Week 2. % % % % % % % % % % % % % % % % % % % % % % % % V e h i c l e c o un t No sharingCTSPCTSPAV,K=1
CTSPAV,K=2
CTSPAV,K=3
CTSPAV,K=4
Figure 12 Total Number of Cars Used on Week 2. reducing vehicle usage and miles traveled, as well as to examine some of the real-world benefitsand drawbacks of the AV ridesharing platform.Figure 11 provides an overview of the trip demand from the dataset and reports the number ofongoing trips for every 15-minute interval throughout the four days considered. The data exhibitsclear and consistent commuting patterns: the inbound demand peaks between 7–8 am, and theoutbound demand peaks at around 5 pm every day. The highly consistent nature of the tripdistributions highlights the opportunities in optimizing them. Part of the results for this case study is obtained by performing further analysis on the results from an earlier work(Hasan and Van Hentenryck in press 2021) which utilized the column-generation heuristic to solve the CTSPAV. asan and Van Hentenryck:
Commuting with Autonomous Vehicles % % % % % % % % % % % % % % % % % % % % % % % % V e h i c l e m il e s t r av e l e d No sharingCTSPCTSPAV,K=1
CTSPAV,K=2
CTSPAV,K=3
CTSPAV,K=4
Figure 13 Total Travel Distance on Week 2.
Figure 12 summarizes results of the primary objective of the CTSPAV for various vehicle capacities K ∈ { , , , } . It reports the total number of vehicles needed to cover all trips for each K valueby aggregating the final vehicle count results of every cluster. The number of vehicles utilizedunder no-sharing conditions (i.e., when commuters travel using their personal vehicles) and underthe original CTSP (with K = 4) (i.e., when drivers are selected from the set of commuters) areincluded for additional perspectives. The percentages in the figure report each count as a fractionof the no-sharing count. The figure highlights the significant capability of the CTSPAV in reducingthe number of vehicles. Indeed, the CTSPAV reduces the vehicle counts by up to 92% every day ,and improves upon the original CTSP by an additional 34%. In fact, the results show that, evenwithout any ride sharing (i.e., when K = 1), AVs still reduce the number of vehicles by 82% andimprove upon the CTSP by an additional 24%. This reduction in vehicle count can be translatedinto a significant reduction in parking spaces, which can then be utilized for other, more useful,infrastructures. The difference in vehicle counts between the CTSP and the CTSPAV is due toautonomy: the vehicles are not associated with drivers and can travel back and forth betweenresidential neighborhoods and workplaces. In the CTSP, vehicles only make a single inbound andoutbound trip every day as their routes are restricted to begin and end at the trip origins anddestinations of their drivers.Figure 13 summarizes the total travel distance of the vehicles, which is the secondary objectiveof the CTSPAV, under the same configurations. The results are again obtained by aggregatingthe results from every cluster and the percentages represent each quantity as a fraction of the no-sharing total. The first result that stands out is how many more miles are traveled by the CTSPAVwhen K = 1 (92–94% more than those under no-sharing conditions). When K = 1 for the CTSPAV,the autonomous vehicles need to perform significantly more back-and-forth traveling between theneighborhoods and the workplace to cover the same amount of trips, which consequently leads to asan and Van Hentenryck: Commuting with Autonomous Vehicles % % % % % % % % % % % % A v er ag e e m p t y m il e s CTSPAV, K=1CTSPAV, K=2CTSPAV, K=3CTSPAV, K=4
Figure 14 Average Empty Miles Per Vehicle on Week 2. . x 2 . x . x 0 . x . x 1 . x R o u t e e ff i c i e n c y (t r i p c o un t / m il e s ) No sharingCTSP
CTSPAV, K=1
CTSPAV, K=2CTSPAV, K=3CTSPAV, K=4
Figure 15 Efficiency of Vehicle Routes their inflated total travel distance. The results improve significantly when K is increased to 2 asthe vehicles allow for more trip aggregations, yet the traveled miles are still 5–6% more than thosefor private vehicles. Net savings in travel distance are only realized when K ≥
3: beyond this point, the reduction in travel distance from ride sharing exceeds the additional empty miles (the milestraveled by an AV with no passengers onboard) introduced by the back-and-forth traveling of theAVs.
Nevertheless, the 29–30% reduction in miles traveled when K = 4 is still not as significant asthat offered by the original CTSP which is around 56–57%. Indeed, the CTSP does not introduceany empty miles and benefits from all the distance savings from ride sharing. On the other hand,the CTSPAV total will necessarily include some empty miles from when the vehicles travel withoutany passengers onboard as they go from the workplace back to the residential neighborhoods inthe morning (or vice versa in the evening) to pick up more trips. There is obviously a tradeoffbetween the reductions in vehicle counts and travel distances. Figure 14 provides a closer lookat the average empty miles per vehicle for the various vehicle capacities. The results are quiteintuitive: the average decreases as K increases, since the larger vehicle capacities allow for moreridesharing and require less back-and-forth traveling to cover the same amount of trips. asan and Van Hentenryck: Commuting with Autonomous Vehicles Figure 15 then attempts to quantify the route efficiency of of the various configurations, i.e.,the number of trips covered per mile traveled. It also includes a multiplicative factor for eachquantity as a multiple of the no-sharing value. The results indicate that the CTSP produces themost efficient routes, whereas the CTSPAV, when K = 1, is the least efficient. The CTSPAV gainsmore efficiency (albeit at a decreasing rate) as its vehicle capacity increases: while its routes aremore efficient than those of the private vehicles when K = 4, they still cannot outperform thoseof the CTSP. There is an intuitive explanation for this observation. The CTSPAV loses its routeefficiency from its empty miles and then has to recover them by maximizing ridesharing to cover asmany trips as possible. In contrast, the CTSP does not have to contend with any efficiency lossesdue to empty miles. Figure 16 presents results on congestion to understand the reduction (or increase) in traffic causedby AVs compared to the no-sharing condition. It tallies the total number of vehicles used by eachconfiguration over every 15-minute interval throughout the four days considered. The goal is toinvestigate, qualitatively and comparatively, the capability of each configuration in flattening thetraffic curve originally produced by the private vehicles. The CTSPAV with K = 1 appears toaggravate traffic as its curve is as tall as, and is wider than, that of private vehicles. This is notsurprising. As illustrated earlier, this configuration produces the largest amount of vehicle milestraveled and also the most empty miles. The curve is drastically flattened as soon as K increasesto 2, and it keeps becoming flatter (at a decreasing rate) as K further increases. When K = 4,the CTSPAV produces about a 60% reduction in traffic. The traffic curves of the CTSP appear todominate slightly those of the CTSPAV with K = 4 most of the time. This observation is also inline with the route efficiency calculations. However, regardless of their relative performance, Figure16 provides evidence that both the CTSP and CTSPAV have the potential to significantly reducetraffic congestion and parking utilization. Figure 17 aims to quantify the relative amount of ride sharing taking place throughout each day forthe different configurations. It reports the average number of riders per vehicle for every 15-minuteinterval throughout the four days considered. Results for the private vehicles and for the CTSPAVwith K = 1 are not included for obvious reasons (they do not allow any sharing). The amount ofride sharing throughout a typical weekday mimics the shape of the trip demand: they both peakduring the same periods of the day. This is to be expected as the CSTSP and CTSPAV maximizeride sharing, which is easier when the trip demand is higher. The figure also shows that the relativeamount of sharing for the CTSPAV increases with vehicle capacity. Moreover, when K = 4 , there asan and Van Hentenryck: Commuting with Autonomous Vehicles CTSPAV, K =1CTSPAV, K =2CTSPAV, K =3CTSPAV, K =4 CTSPNo sharing V e h i c l e c o un t Hour of day (a) Monday
No sharing
CTSPCTSPAV, K =4 CTSPAV, K =3 CTSPAV, K =2 CTSPAV, K =10 V e h i c l e c o un t Hour of day (b) Tuesday
No sharing
CTSP
CTSPAV, K =4CTSPAV, K =3 CTSPAV, K =2CTSPAV, K =1 V e h i c l e c o un t Hour of day (c) Wednesday
No sharing CTSPCTSPAV, K =4 CTSPAV, K =3 CTSPAV, K =2CTSPAV, K =10 V e h i c l e c o un t Hour of day (d) Thursday
Figure 16 Number of Vehicles on the Road Over 15-Minute Intervals on Week 2. is more ride sharing in the CTSPAV than in the CTSP most of the time.
This can be attributed tothe relative flexibility of the mini routes of the CTSPAV compared to those of the CTSP. Indeed,a CTSP route must start and end at the orign and destination of its driver, which constrains itstotal duration by the ride-duration constraints on its driver. Mini routes of the CTSPAV are notsubjected to these restrictions, allowing for more flexibility in serving trips. Interestingly, duringpeaks, the average amount of ride sharing is between 3.0 and 3.5 due to the spatial and temporalproperties of the commuting trips. This also indicates the types of autonmous vehicles that will bemost useful in the future, at least for cities like Ann Arbor.Figure 18 reports the average commute times, i.e., the average time spent on the vehicle by eachrider. The percentages of each quantity are calculated relative to the no-sharing value. The resultsshed light on another inherent trade-off in ride-sharing service as the ride duration necessarilyincraeses. During ridesharing, a route may deviate from the optimal path to pickup or drop offother riders. This, combined with possible wait times incurred at the pickup locations, contributeto the increased ride duration. The results reveal an expected trend for the CTSPAV: the averagecommute times increase with an increase in vehicle capacity. However, it is interesting to observethat, although parameter R was set to 50% for the case study, the commute times of the CTSPAVwith K = 4 only increase by an average of 26%. The CTSPAV thus guarantees a high quality ofservice for its riders. asan and Van Hentenryck: Commuting with Autonomous Vehicles
20 21 A v er ag e r i d er s p er v e h i c l e Hour of day
CTSP
CTSPAV, K=3 CTSPAV, K=4
CTSPAV, K=2 (a) Monday
20 21 A v er ag e r i d er s p er v e h i c l e Hour of day
CTSP
CTSPAV, K=3 CTSPAV, K=4
CTSPAV, K=2 (b) Tuesday
20 21 A v er ag e r i d er s p er v e h i c l e Hour of day
CTSP
CTSPAV, K=3 CTSPAV, K=4
CTSPAV, K=2 (c) Wednesday
20 21 A v er ag e r i d er s p er v e h i c l e Hour of day
CTSP
CTSPAV, K=3 CTSPAV, K=4
CTSPAV, K=2 (d) Thursday
Figure 17 Average Riders Per Vehicle Over 15-Minute Intervals on Week 2 % % % % % % % % % % % % % % % % % % % % % % % % A v er ag e c o mm u t e t i m e ( m i n s ) No sharingCTSPCTSPAV,K=1
CTSPAV,K=2
CTSPAV,K=3
CTSPAV , K=4
Figure 18 Average Commute Time on Week 2.
9. Conclusion
The purpose of the CTSPAV is to synthesize an optimal routing plan for serving a large set ofcommute trips with AVs. Its design was originally motivated by the desire to address the growingparking and traffic congestion problems induced by the average of 9,000 daily commuters travelingto parking lots operated by the University of Michigan located in downtown Ann Arbor, Michigan.Utilization of AVs was seen as the key to addressing the shortcomings of the original CTSP—aconventional car-pooling problem with the same objectives as the CTSPAV—by obviating anydriver-related requirements that could limit its ridesharing potential. A first attempt at solving theproblem by Hasan and Van Hentenryck (in press 2021) investigated two different methods: (1) ACTSPAV procedure which used column-generation to discover mini routes—short routes covering asan and Van Hentenryck:
Commuting with Autonomous Vehicles only inbound or outbound trips that have distinct pickup, transit, and drop-off phases—withnegative reduced costs which are chained together to form longer AV routes in its master problemand (2) A DARP procedure which uses a classical column-generation approach originally developedfor the DARP to solve the CTSPAV. Both methods utilized identical lexicographic objectives whichsought to first minimize the required vehicle count and then minimize their total travel distance.To deal with the complexity of handling the massive volume of trips, the commuters were firstclustered into groups representing artificial neighborhoods, after which ridesharing within eachcluster was optimized exclusively. They discovered that each method had a trade-off: The CTSPAVprocedure produced strong integer solutions but had weak primal lower bounds. Conversely, theDARP procedure generated stonger primal lower bounds especially for the primary objective, but itwas slow and therefore could not obtain strong integer solutions within time-constrained scenarios. The trade-offs of the two procedures presented an opportunity for exploring a method that couldleverage the strengths of both, which is the primary methodological contribution of this work.
Thispaper thus proposed a branch-and-cut procedure that exploits a dual-modeling approach for solvingthe CTSPAV. The core of the procedure is a MIP formulation of the CTSPAV that chains (exhaus-tively enumerated) mini routes to form longer AV routes and is capable of producing high-qualityinteger solutions quality. This core is complemented by a DARP formulation whose relaxation(for minimizing vehicle counts) is obtained through a column-generation procedure. The DARPformulation is less effective in finding high-quality integer solution, but its relaxation producesstronger lower bounds. The overall algorithm solves the core branch-and-cut procedure and theDARP relaxation in parallel, transmitting new lower bounds asynchrously from the relaxation tothe branch and cut procedure. Computational evaluations that use instances derived from theAnn Arbor commute-trip data demonstrated that this hybrid algorithm consistently outperformsa similar branch-and-cut procedure that utilizes other well-established valid inequalities like 2-pathcuts and successor and predecessor inequalities. It also successfully closes the optimality gaps forseveral large and medium-sized instances as well as those for all tight problem instances consideredin the evaluation, of which none could be optimally solved by the CTSPAV procedure of Hasanand Van Hentenryck (in press 2021).With the availabilty of an exact branch and cut procedure, the paper then provided a comprehen-sive analysis of the potential of AVs for ride-sharing platforms and relieving parking pressure andcongestion in medium-sized cities. In particular, the paper presented results of a case study whichapplies the clustering-CTSPAV optimization workflow on a large-scale dataset of commute tripsfrom the city of Ann Arbor, Michigan.
The analysis revealed several invaluable insights, includ-ing the CTSPAV capability of reducing daily vehicle counts by 92%, further improving upon thealready massive 57% vehicle reductions of the original CTSP . It does so by generating AV routes asan and Van Hentenryck:
Commuting with Autonomous Vehicles that are very long—a stark contrast to the short routes of the CTSP—allowing each AV to coversignificantly more trips every day. It could also effectively flatten the vehicle usage curve (i.e.,the number of vehicles used per unit time), suggesting a concomitant ability to effectively reducetraffic congestion. The CTSPAV also produced higher averages for trips shared per unit time thanthe CTSP, indicating that it is superior at aggregating more trips for ridesharing. The analysisalso revealed some drawbacks, the most significant being the introduction of empty miles into thedaily travel distance totals. The empty miles degrade the efficiency of the CTSPAV routes, whichmeasures the average number of trips covered per distance traveled, making them less efficientthan the routes of the CTSP. Empty miles are unfortunately a by-product that is inherent to theutilization of AVs, and its introduction is a trade-off that will need to be carefully weighed againstthe benefits of AVs by the ridesharing platform operator. Nonetheless, the results indicate that theCTSPAV routing plan, even with its empty miles, is still able to reduce the total miles traveledby private vehicles by 30% while producing routes that at 1.4 times more efficient. On the whole,the case study shows that a CTSPAV-based ridesharing platform could significantly reduce dailyvehicle counts, as well as the number of vehicles used per unit time. Such a platform would behighly effective at aggregating trips, making it a very promising solution for reducing parking spaceutilization and for mitigating traffic congestion induced by large-scale commuting. Acknowledgments
We would like to thank Stephen Dolen from Logistics, Transportation, and Parking of the University ofMichigan for his assistance in obtaining the dataset used in this research. Part of this research was funded bythe Rackham Graduate Student Research Grant, computational resources and services provided by AdvancedResearch Computing at the University of Michigan, NSF Leap HI proposal NSF-1854684, and Departmentof Energy Research Award 7F-30154.
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Sustainable Cities and Society http://dx.doi.org/https://doi.org/10.1016/j.scs.2015.07.006 . asan and Van Hentenryck: Commuting with Autonomous Vehicles Appendix. Filtering of Graph G Graph G can be made more compact by only retaining edges that satisfy a priori route-feasibility con-straints. This is done by pre-processing time-window, pairing, precedence, and ride-duration limit constraintson A to identify and eliminate edges that are infeasible, i.e., those that cannot belong to any feasible AVroute. In this work, the set of infeasible edges is identified using a combination of rules proposed by Dumaset al. (1991) and Cordeau (2006). These rules are presented in the Appendix.(a) Direct trips to and from the depot: • { ( v s , v t ) , ( v t , v s ) } • { ( i, v s ) , ( i, v t ) , ( v t , i ) : i ∈ P} • { ( v s , i ) , ( i, v s ) , ( v t , i ) : i ∈ D} (b) Precedence of pickup and drop-off nodes of inbound and outbound trips of each commuter (constraints(14)): { ( i, n + i ) , ( i, n + i ) , ( n + i, i ) , ( n + i, n + i ) , (2 n + i, i ) , (2 n + i, n + i ) , (3 n + i, i ) , (3 n + i, n + i ) , (3 n + i, n + i ) : i ∈ P + } (c) Precedence of pickup and drop-off nodes of inbound and outbound mini routes: • { ( i, j ) : i ∈ P + ∧ j ∈ P − ∪ D − } • { ( i, j ) : i ∈ D + ∧ j ∈ D − } • { ( i, j ) : i ∈ P − ∧ j ∈ P + ∪ D + } • { ( i, j ) : i ∈ D − ∧ j ∈ D + } (d) Time windows along each edge: { ( i, j ) : ( i, j ) ∈ A \ { δ + ( v s ) ∪ δ − ( v t ) } ∧ a i + s i + τ ( i,j ) > b j } (e) Ride-duration limit of each commuter: { ( i, j ) , ( j, n + i ) : i ∈ P ∧ j ∈ P ∪ D ∧ i (cid:54) = j ∧ τ ( i,j ) + s j + τ ( j,n + i ) >L i } (f) Time windows and ride-duration limits of pairs of trips: • { ( i, n + j ) : i, j ∈ P ∧ i (cid:54) = j ∧ ¬ f easible ( j → i → n + j → n + i ) } • { ( n + i, j ) : i, j ∈ P ∧ i (cid:54) = j ∧ ¬ f easible ( i → n + i → j → n + j ) } • { ( i, j ) : i, j ∈ P ∧ i (cid:54) = j ∧ ¬ f easible ( i → j → n + i → n + j ) ∧ ¬ f easible ( i → j → n + j → n + i ) } • { ( n + i, n + j ) : i, j ∈ P ∧ i (cid:54) = j ∧ ¬ f easible ( i → j → n + i → n + j ) ∧ ¬ f easible ( j → i → n + i → n + j ) } Note that the sets of edges in (f) utilize the f easible function to determine if a partial route satisfies time-window and ride-duration limit constraints. For instance, the first condition indicates that edge ( i, n + j ) isinfeasible if the partial route j → i → n + j → n + i is infeasible. Figure 19 illustrates an example of graph G resulting from this pre-processing step. asan and Van Hentenryck: Commuting with Autonomous Vehicles Inbound Route Graph,
For each commuter i : • Origin node: i • Destination node: n + i Virtual source node: 0Virtual sink node: 2 n + 1 Figure 19 Graph G (Each Dotted Line Represents a Pair of Bidirectional Edges). Appendix. Computational Results
Table 3 summarizes the results of CTSPAV
Hybrid for every large problem instance. Its first column showsthe name of every instance. The next three columns display properties that characterize the size of eachinstance. They list the node count of graph G , |N | , the edge count of the graph (after the pre-processingstep), |A| , and finally the number of mini routes generated by the MREA, | Ω | , for every instance. The nextcolumn shows the wall time spent to the enumerate the mini routes. The remaining columns summarizethe results of CTSPAV Hybrid . The first two show the vehicle count and total travel distance from its bestincumbent solution. The next two display the absolute gap for the vehicle count and the optimality gap forthe objective value of the best incumbent solution. The following column shows the number of tree nodesexplored in the solution process. The last two columns display the (total) wall time spent to solve the MIPand that spent to close the vehicle count gap. For the very last column, values are only listed for instanceswhereby the vehicle count gap could be closed within the 2-hour time limit. It is left blank otherwise. Tables5 and 7 provide the same set of information for CTSPAV
Hybrid for every medium and tight problem instancerespectively. On the other hand, Tables 4, 6, and 8 show the results of CTSPAV
SEC and CTSPAV
Base for alllarge, medium, and tight problem instances respectively.Tables 9, 10, and 11 list the heuristic results for every large, medium, and tight instance respectively. Theirfirst columns show the instance names, followed by three columns that show the number of columns (miniroutes) generated, the final vehicle count, and the total travel distance for every instance. The following twocolumns display the absolute gap of its vehicle count results and the optimality gap of its best incumbentsolution. Since the heuristic does not utilize all feasible mini routes, it has to use the optimal LP-relaxationsolution of RMP
CTSPAV to derive primal lower bounds for these gap calculations. The final three columnsshow the percentage difference between the column count, the vehicle count, and the total distance of theheuristic relative to those of CTSPAV
Hybrid . asan and Van Hentenryck: Commuting with Autonomous Vehicles Table 3 Results of CTSPAV
Hybrid for the Large Problem Instances
Instancename Nodecount Edgecount Miniroutecount Routeenumerationtime (s) Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s)MIP OptimalcountL0 402 23983 3730 22 3 642049 0 0.0 156016 5360 1284L1 402 22621 1093 21 3 463065 1 33.3 524584 7200 -L2 402 26781 51175 24 4 817348 2 49.9 6424 7200 -L3 402 26496 63597 24 4 841180 2 49.9 7430 7202 -L4 402 25309 49147 23 4 813018 1 24.9 11734 7201 -L5 402 22425 1605 20 3 512675 1 33.3 189596 7200 -L6 402 26420 20060 23 4 955285 2 49.9 7935 7201 -L7 402 24699 21403 23 4 888490 1 24.9 22067 7201 -L8 402 25710 14818 23 4 844674 1 24.9 23822 7200 -L9 402 27315 191067 25 5 737361 3 59.9 1511 7200 -L10 402 24386 5807 25 3 555102 1 33.3 30016 7201 -L11 402 25639 18237 23 3 570036 1 33.3 13176 7201 -L12 402 23748 3631 21 3 581863 1 33.3 125059 7200 -L13 402 24581 6835 24 3 624843 1 33.3 23394 7202 -L14 402 26287 72200 23 4 949361 2 49.9 5138 7201 -L15 402 24898 114817 38 4 1108007 2 49.9 7258 7200 -L16 402 24203 9231 22 4 847394 1 24.9 75500 7200 -L17 402 23734 6404 22 4 863265 0 0.0 22485 7200 5883L18 402 24712 4417 33 4 914762 1 24.9 33188 7201 -L19 402 25513 35873 24 3 698599 1 33.3 11984 7201 -L20 402 25528 58833 23 3 779684 1 33.3 8639 7200 -L21 402 22832 4870 21 2 457911 0 0.0 166142 7200 2217
Table 4 Results of CTSPAV
SEC and CTSPAV
Base for the Large Problem Instances
Instancename CTSPAV variantSEC BaseVehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s) Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s)MIP Optimalcount MIP OptimalcountL0 3 646884 1 33.3 43103 7200 - 3 652906 2 66.5 24638 7201 -L1 3 463065 1 33.3 135613 7228 - 3 463065 2 66.6 408157 7202 -L2 4 821989 2 49.9 6369 7218 - 4 824321 3 74.8 5229 7201 -L3 4 849844 2 49.9 4713 7215 - 4 843208 3 74.8 5291 7200 -L4 5 820800 3 59.9 10005 7202 - 5 831319 3 59.9 20952 7201 -L5 3 512838 1 33.3 73463 7202 - 3 512675 2 66.5 195089 7201 -L6 4 971911 2 49.9 9541 7207 - 4 967746 3 74.8 11540 7204 -L7 4 891808 2 49.9 7244 7206 - 4 893550 3 74.8 15275 7201 -L8 4 845333 2 49.9 8301 7201 - 4 845100 3 74.8 16814 7200 -L9 5 730915 3 59.9 2023 7200 - 5 720023 4 79.9 1906 7200 -L10 3 555102 1 33.3 21162 7200 - 3 555102 2 66.5 21223 7201 -L11 3 573246 1 33.3 3428 7203 - 3 574227 2 66.5 21195 7200 -L12 3 581863 1 33.3 34193 7200 - 3 581863 2 66.5 57588 7202 -L13 3 626100 1 33.3 15871 7221 - 3 625042 2 66.5 36251 7201 -L14 4 949659 2 49.9 5431 7213 - 4 932389 3 74.8 4986 7200 -L15 4 1108620 2 49.9 4435 7202 - 4 1116187 3 74.8 2732 7201 -L16 4 857161 2 49.9 12595 7203 - 4 846684 3 74.8 21489 7200 -L17 4 867674 2 49.9 21259 7201 - 4 865011 2 49.9 21691 7200 -L18 4 917395 2 49.9 18251 7201 - 4 914762 2 49.9 20825 7200 -L19 4 697540 2 49.9 4925 7298 - 4 706887 3 74.9 15757 7200 -L20 3 772418 1 33.3 6277 7318 - 3 778248 2 66.5 7573 7200 -L21 3 447435 2 66.6 1632 7259 - 2 458460 1 49.9 86453 7205 - asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 5 Results of CTSPAV
Hybrid for the Medium Problem Instances
Instancename Nodecount Edgecount Miniroutecount Routeenumerationtime (s) Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s)MIP OptimalcountM0 302 14024 3233 7 2 481141 0 0.0 109840 7200 445M1 262 11267 8986 6 3 605515 1 33.3 39142 7200 -M2 302 13973 31559 7 3 847030 0 0.0 27300 7200 4567M3 302 15253 30739 10 3 668490 1 33.3 18968 7201 -M4 302 14426 28359 9 3 535195 1 33.3 19036 7201 -M5 302 12739 503 6 2 333048 0 0.0 1348803 3409 340M6 302 15515 47521 8 3 657988 1 33.3 12023 7200 -M7 302 14485 3485 7 3 595519 1 33.3 123341 7200 -M8 302 15404 10828 8 3 689147 1 33.3 21890 7201 -M9 302 15882 55026 9 3 489997 1 33.3 14828 7201 -M10 302 14898 119198 10 3 719639 1 33.3 18473 7200 -M11 302 13800 5845 10 2 602968 0 0.0 205444 7200 1814M12 302 13542 1884 7 2 417175 0 0.0 61043 1007 122M13 302 14564 28922 9 3 652724 1 33.3 18510 7200 -M14 302 13902 3207 7 2 401064 0 0.0 51325 2406 270M15 302 14801 14693 7 3 627967 0 0.0 39332 7200 7030M16 254 10233 3968 4 3 599126 0 0.0 30465 2949 2787M17 302 13224 1380 7 2 490178 0 0.0 14669 134 73M18 290 11758 749 5 2 347259 0 0.0 30780 418 416M19 302 13043 3174 7 2 339073 0 0.0 278853 6566 6004M20 302 14184 4380 7 3 551547 1 33.3 81164 7200 -M21 258 10135 1696 6 3 620764 0 0.0 273752 4256 4116M22 302 14856 19435 8 3 683612 1 33.3 18247 7200 -M23 302 14230 12339 7 3 556522 1 33.3 31373 7200 -M24 302 14694 23970 7 3 588191 1 33.3 18586 7200 -M25 286 13139 19056 6 3 596412 1 33.3 24223 7201 -M26 302 13505 1547 11 3 445952 0 0.0 55454 1576 1311M27 262 10980 4981 4 3 712881 0 0.0 34804 1648 1422M28 302 13883 3565 11 2 394323 0 0.0 1737 183 104M29 302 15142 38021 10 3 729149 1 33.3 18677 7200 - asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 6 Results of CTSPAV
SEC and CTSPAV
Base for the Medium Problem Instances
Instancename CTSPAV variantSEC BaseVehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s) Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s)MIP Optimalcount MIP OptimalcountM0 2 480223 0 0.0 229608 7200 2756 2 480225 1 49.9 143747 7201 -M1 3 603771 1 33.3 21394 7203 - 3 604103 2 66.5 20833 7200 -M2 3 846579 1 33.2 16221 7207 - 3 846597 1 33.2 21540 7201 -M3 3 668248 1 33.3 15377 7205 - 3 682726 2 66.5 21125 7200 -M4 3 535334 1 33.3 7076 7298 - 3 535195 2 66.5 21423 7200 -M5 2 333048 0 0.0 14122 335 95 2 333366 1 49.9 1119738 7200 -M6 3 656983 1 33.3 6422 7201 - 4 655969 3 74.9 20905 7200 -M7 3 595519 1 33.3 45152 7204 - 3 595519 2 66.5 65095 7201 -M8 3 679167 1 33.3 21493 7201 - 3 687498 2 66.5 21259 7200 -M9 3 489461 1 33.3 5734 7238 - 3 497878 2 66.6 21032 7200 -M10 3 719788 1 33.3 8781 7215 - 3 722278 2 66.5 3697 7201 -M11 2 601111 0 0.0 35354 7202 1730 2 601041 1 49.8 29943 7200 -M12 2 417175 0 0.0 50401 1911 195 2 417185 1 49.9 251358 7200 -M13 3 655996 1 33.3 7966 7212 - 3 653183 2 66.5 21185 7202 -M14 2 401064 0 0.0 26183 5314 983 2 401064 1 49.9 92983 7200 -M15 4 622760 2 49.9 20584 7203 - 4 622717 2 49.9 23019 7200 -M16 3 599126 1 33.3 80256 7205 - 3 599442 2 66.5 32695 7200 -M17 2 490178 0 0.0 4120 141 58 2 490178 1 49.9 272323 7200 -M18 2 347259 0 0.0 436 156 151 2 347259 1 49.9 1064235 7201 -M19 2 339073 0 0.0 4695 1645 637 2 339073 1 49.9 192573 7200 -M20 3 551547 1 33.3 41920 7203 - 3 551547 2 66.5 39175 7200 -M21 3 620783 1 33.3 211984 7200 - 3 620764 2 66.5 319796 7200 -M22 3 685043 1 33.3 15662 7205 - 3 683300 1 33.3 24972 7200 -M23 3 556571 1 33.3 21292 7200 - 3 555996 2 66.5 20915 7200 -M24 3 588191 1 33.3 17174 7223 - 3 587860 2 66.5 21374 7200 -M25 3 597367 1 33.3 20807 7200 - 3 596653 2 66.5 21527 7201 -M26 3 445952 1 33.3 114827 7202 - 3 445952 2 66.6 189359 7200 -M27 3 712881 1 33.3 25439 7202 - 3 712881 2 66.5 21500 7200 -M28 2 394323 0 0.0 1830 431 241 2 394323 1 49.9 139970 7200 -M29 3 731148 1 33.3 10958 7204 - 3 729946 2 66.5 19975 7203 - asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 7 Results of CTSPAV
Hybrid for the Tight Problem Instances
Instancename Nodecount Edgecount Miniroutecount Routeenumerationtime (s) Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s)MIP OptimalcountS0 402 20870 374 19 5 961566 0 0.0 144186 544 129S1 402 20847 267 18 3 619257 0 0.0 19909 143 124S2 402 21424 971 20 5 1246019 0 0.0 27515 459 333S3 402 21472 1268 21 5 1192722 0 0.0 19049 830 721S4 402 21352 1204 20 5 1187914 0 0.0 957524 5084 238S5 402 20918 304 17 3 676142 0 0.0 1887 28 24S6 402 21050 707 20 6 1503404 0 0.0 14494 224 187S7 402 21022 687 20 5 1345009 0 0.0 121198 1524 1180S8 402 20896 581 31 5 1310231 0 0.0 2705 37 32S9 402 21876 1666 30 6 1094536 0 0.0 14475 384 262S10 402 21044 430 29 4 805606 0 0.0 17905 228 228S11 402 21614 835 29 4 819652 0 0.0 11194 211 188S12 402 20946 393 32 4 837723 0 0.0 448878 1504 86S13 402 21137 504 20 4 914708 0 0.0 136179 1149 667S14 402 21438 1056 32 5 1450697 0 0.0 10064 71 17S15 402 21156 2825 31 5 1613836 0 0.0 2646 20 8S16 402 21005 528 32 5 1220586 0 0.0 8396 147 136S17 402 20844 499 30 5 1252397 0 0.0 9523 68 34S18 402 20713 392 31 6 1452716 0 0.0 18044 201 200S19 402 21377 1267 31 4 1030225 0 0.0 513121 4069 1218S20 402 21542 1541 33 4 1144849 0 0.0 8369 222 211S21 402 20959 313 19 3 580008 0 0.0 204235 2267 2131
Table 8 Results of CTSPAV
SEC and CTSPAV
Base for the Tight Problem Instances
Instancename CTSPAV variantSEC BaseVehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s) Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Nodesexplored Wall time (s)MIP Optimalcount MIP OptimalcountS0 5 961566 0 0.0 95266 388 82 5 961566 0 0.0 151291 533 90S1 3 619257 0 0.0 9643 97 89 3 619257 0 0.0 24230 326 323S2 5 1246019 0 0.0 13952 277 203 5 1246019 0 0.0 21299 917 902S3 5 1192722 1 20.0 178946 7201 - 5 1192722 1 19.9 241540 7201 -S4 5 1187914 0 0.0 400941 2668 187 5 1187914 0 0.0 17315 406 225S5 3 676142 0 0.0 3023 13 5 3 676142 0 0.0 4393 14 6S6 6 1503404 0 0.0 14190 284 284 6 1503404 0 0.0 73967 1653 1567S7 5 1345009 0 0.0 216353 3000 2352 5 1345009 0 0.0 243780 2824 1953S8 5 1310231 0 0.0 1459 22 21 5 1310231 0 0.0 3948 46 44S9 6 1094536 1 16.6 152214 7202 - 6 1094536 1 16.6 193079 7201 -S10 4 805606 0 0.0 16966 236 226 4 805606 0 0.0 9222 84 80S11 4 819652 0 0.0 9997 168 150 4 819652 0 0.0 12619 210 197S12 4 837723 0 0.0 161991 665 99 4 837723 0 0.0 155274 554 157S13 4 914708 0 0.0 94311 1553 1301 4 914708 0 0.0 304917 5579 5231S14 5 1450697 0 0.0 1250 44 31 5 1450697 0 0.0 14449 39 7S15 5 1613836 0 0.0 3338 24 12 5 1613836 0 0.0 1600 19 11S16 5 1220586 0 0.0 3471 78 72 5 1220586 0 0.0 1348 54 52S17 5 1252397 0 0.0 7338 48 32 5 1252397 0 0.0 10428 76 55S18 6 1452716 0 0.0 26594 278 268 6 1452716 0 0.0 22340 375 374S19 4 1030225 0 0.0 553629 4371 1324 4 1030225 0 0.0 173744 1736 326S20 4 1144849 0 0.0 9894 199 188 4 1144849 0 0.0 22515 504 500S21 3 580008 0 0.0 138098 2078 2042 3 580008 1 33.3 886186 7201 - asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 9 Results of CTSPAV Column-Generation Heuristic by Hasan and Van Hentenryck (in press 2021) forLarge Problem Instances
Instancename Columncount Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Percentage differenceColumncount Vehiclecount TotaldistanceL0 2231 3 647661 2 66.5 -40% 0% -0.75%L1 901 3 463065 2 66.6 -18% 0% 0.00%L2 8713 4 817348 3 74.9 -83% 0% 0.05%L3 9347 4 841180 3 74.8 -85% 0% -0.30%L4 7253 4 813018 3 74.8 -85% 0% -0.14%L5 960 3 512675 2 66.6 -40% 0% 0.00%L6 6330 4 955285 3 74.8 -68% 0% 1.19%L7 5087 4 888490 3 74.8 -76% 0% 0.35%L8 4902 4 844674 3 74.8 -67% 0% 0.00%L9 13892 5 737361 4 79.9 -93% 0% -0.22%L10 2884 3 555102 2 66.5 -50% 0% 0.05%L11 5659 3 570036 2 66.5 -69% 0% 0.88%L12 2116 3 581863 2 66.5 -42% 0% 0.00%L13 3106 3 624843 2 66.5 -55% 0% 0.01%L14 9539 4 949361 3 74.8 -87% 0% -0.66%L15 8161 4 1108007 3 74.8 -93% 0% 0.80%L16 3513 4 847394 3 74.8 -62% 0% 0.37%L17 2886 4 862155 3 74.8 -55% 0% 0.11%L18 2912 4 914762 3 74.8 -34% 0% 0.49%L19 6278 3 698599 2 74.9 -82% 33% 0.27%L20 8291 3 779684 2 66.5 -86% 0% -1.40%L21 1397 2 457911 1 49.9 -71% 0% -0.01% asan and Van Hentenryck:
Commuting with Autonomous Vehicles Table 10 Results of CTSPAV Column-Generation Heuristic by Hasan and Van Hentenryck (in press 2021) forMedium Problem Instances
Instancename Columncount Vehiclecount Totaldistance(m) Vehiclecountgap Optimalitygap (%) Percentage differenceColumncount Vehiclecount TotaldistanceM0 1664 2 481141 1 49.9 -49% 0% -0.19%M1 2643 3 605515 2 66.5 -71% 0% -0.17%M2 4349 3 846579 2 66.5 -86% 0% 0.75%M3 5461 3 668490 2 66.5 -82% 0% 1.07%M4 3556 3 535195 2 66.6 -87% 0% 0.04%M5 464 2 333048 1 49.9 -8% 0% 0.00%M6 6217 3 657988 2 66.5 -87% 0% 0.36%M7 2081 3 595519 2 66.5 -40% 0% 0.00%M8 3728 3 689147 2 66.5 -66% 0% -0.21%M9 6545 3 489997 2 66.6 -88% 0% 0.01%M10 6938 3 719639 2 66.5 -94% 0% 0.03%M11 2142 2 602968 1 49.9 -63% 0% -0.40%M12 1198 2 417175 1 49.9 -36% 0% 0.00%M13 4821 3 652724 2 66.5 -83% 0% 0.17%M14 1712 2 401064 1 49.9 -47% 0% 0.08%M15 4122 3 627967 2 74.9 -72% 33% -1.07%M16 1849 3 599126 2 66.5 -53% 0% 0.07%M17 964 2 490178 1 49.9 -30% 0% 0.00%M18 528 2 347259 1 49.9 -30% 0% 0.00%M19 914 2 339073 1 49.9 -71% 0% 0.00%M20 2153 3 551547 2 66.5 -51% 0% 0.00%M21 1172 3 620764 2 66.5 -31% 0% 0.00%M22 4527 3 683612 2 66.5 -77% 0% 0.16%M23 3416 3 556522 2 66.5 -72% 0% -0.09%M24 4949 3 588191 2 66.5 -79% 0% 0.04%M25 3969 3 596412 2 66.5 -79% 0% 0.16%M26 1043 3 445952 2 66.6 -33% 0% 0.09%M27 2336 3 712881 2 66.5 -53% 0% 0.02%M28 1810 2 394323 1 49.9 -49% 0% 0.00%M29 6028 3 729149 2 66.5 -84% 0% -0.38% asan and Van Hentenryck: